GNI Foundation

GNI Foundation Evangelism - to educate,
Development - to help the helpless

14/04/2026

๐—ข๐—ก๐—˜ ๐—ข๐—™๐—™๐—œ๐—–๐—˜, ๐—ง๐—›๐—ฅ๐—˜๐—˜ ๐—™๐—จ๐—ก๐—–๐—ง๐—œ๐—ข๐—ก๐—ฆ
๐—” ๐—ง๐—ต๐—ฒ๐—ผ๐—น๐—ผ๐—ด๐—ถ๐—ฐ๐—ฎ๐—น ๐—ฆ๐˜๐˜‚๐—ฑ๐˜† ๐—ผ๐—ป ๐˜๐—ต๐—ฒ ๐—จ๐—ป๐—ถ๐˜๐˜† ๐—ฎ๐—ป๐—ฑ ๐——๐—ถ๐˜€๐˜๐—ฟ๐—ถ๐—ฏ๐˜‚๐˜๐—ถ๐—ผ๐—ป ๐—ผ๐—ณ ๐—–๐—ต๐˜‚๐—ฟ๐—ฐ๐—ต Leadership (Pastor, Elder, and Bishop in Biblical Governance)

๐—œ. ๐—ฃ๐—ฅ๐—ข๐—•๐—Ÿ๐—˜๐—  ๐—ฆ๐—ง๐—”๐—ง๐—˜๐— ๐—˜๐—ก๐—ง

Can pastor, elder, and bishop be three different persons?

๐Ÿ‘‰ Practically: YES โ€” within certain church systems and administrative structures
๐Ÿ‘‰ Biblically: NO โ€” not as three separate and independent offices

Instead, Scripture consistently reveals:

โ€ข One unified leadership office
โ€ข Expressed through multiple functions and dimensions

This leads to a critical distinction:

๐Ÿ‘‰ The OFFICE is singular
๐Ÿ‘‰ The FUNCTIONS may be plural

Failure to distinguish these leads to doctrinal confusion and structural imbalance.

๐—œ๐—œ. ๐—•๐—œ๐—•๐—Ÿ๐—œ๐—–๐—”๐—Ÿ ๐—ฃ๐—ข๐—ฆ๐—œ๐—ง๐—œ๐—ข๐—ก (๐—™๐—ข๐—จ๐—ก๐——๐—”๐—ง๐—œ๐—ข๐—ก๐—”๐—Ÿ ๐——๐—ข๐—–๐—ง๐—ฅ๐—œ๐—ก๐—˜)

The New Testament provides a unified portrait of church leadership through key passages:

โ€ข Acts 20:17โ€“28
โ€ข Titus 1:5โ€“7
โ€ข 1 Peter 5:1โ€“2

From these texts emerges a clear doctrinal synthesis:

๐Ÿ‘‰ Elder = Overseer (Bishop) = Pastor (Shepherd)

These are not ranks, but complementary descriptions of the same office.

A. LEXICAL UNITY (WORD STUDY)

Each term emphasizes a different dimension of the same leader:

โ€ข ฯ€ฮฟฮนฮผฮฎฮฝ (poimฤ“n) โ€” Pastor / Shepherd
โ†’ Emphasizes care, feeding, nurturing

โ€ข ฯ€ฯฮตฯƒฮฒฯฯ„ฮตฯฮฟฯ‚ (presbyteros) โ€” Elder
โ†’ Emphasizes maturity, character, spiritual authority

โ€ข แผฯ€ฮฏฯƒฮบฮฟฯ€ฮฟฯ‚ (episkopos) โ€” Bishop / Overseer
โ†’ Emphasizes oversight, governance, supervision

๐Ÿ‘‰ Not three identities, but one identity described in three ways

B. TEXTUAL SYNTHESIS (EXEGETICAL HARMONY)
Acts 20:17โ€“28

Paul calls the elders (v.17), then describes them as:

โ€ข Overseers (v.28)
โ€ข Who must shepherd the church (v.28)

๐Ÿ‘‰ Same group
๐Ÿ‘‰ Three terms
๐Ÿ‘‰ One office

Titus 1:5โ€“7

โ€ข Elders are appointed (v.5)
โ€ข Then described as bishops (v.7)

๐Ÿ‘‰ No transition of office
๐Ÿ‘‰ Only change of terminology

1 Peter 5:1โ€“2

Peter exhorts:

โ€ข Elders (identity)
โ€ข To shepherd (function)
โ€ข Exercising oversight (authority)

๐Ÿ‘‰ Identity + Function + Authority unified in one role

๐—œ๐—œ๐—œ. ๐—ง๐—›๐—˜๐—ข๐—Ÿ๐—ข๐—š๐—œ๐—–๐—”๐—Ÿ ๐—™๐—ฅ๐—”๐— ๐—˜๐—ช๐—ข๐—ฅ๐—ž (๐—ฆ๐—ฌ๐—ฆ๐—ง๐—˜๐— ๐—”๐—ง๐—œ๐—– ๐—ฆ๐—ฌ๐—ก๐—ง๐—›๐—˜๐—ฆ๐—œ๐—ฆ)
___________________________________
Dimension | Term | Emphasis
___________________________________
Identity | Elder | Who the leader is
Function | Pastor | How the leader serves
Authority | Bishop | What the leader oversees

๐Ÿ‘‰ ONE OFFICE โ€” THREE DIMENSIONS

This is not a hierarchy but a holistic calling.

๐—œ๐—ฉ. ๐—ช๐—›๐—”๐—ง ๐—ฌ๐—ข๐—จ ๐—–๐—”๐—ก ๐——๐—ข (๐—•๐—œ๐—•๐—Ÿ๐—œ๐—–๐—”๐—Ÿ๐—Ÿ๐—ฌ ๐—ฆ๐—ข๐—จ๐—ก๐—— ๐—”๐—ฃ๐—ฃ๐—Ÿ๐—œ๐—–๐—”๐—ง๐—œ๐—ข๐—ก)

Scripture allows functional distribution, not ontological division.

Healthy Model (Plural Leadership)

A church may have:

โ€ข Multiple elders
โ€ข One elder focused on teaching/preaching (commonly called โ€œpastorโ€)
โ€ข Others focusing on governance, discipleship, or administration

๐Ÿ‘‰ All remain:

Elders in identity
Pastors in care
Overseers in responsibility

๐Ÿ‘‰ Differences are functional, not hierarchical

๐—ฉ. ๐—ช๐—›๐—”๐—ง ๐—ฌ๐—ข๐—จ ๐— ๐—จ๐—ฆ๐—ง ๐—”๐—ฉ๐—ข๐—œ๐—— (๐—˜๐—–๐—–๐—Ÿ๐—˜๐—ฆ๐—œ๐—ข๐—Ÿ๐—ข๐—š๐—œ๐—–๐—”๐—Ÿ ๐—˜๐—ฅ๐—ฅ๐—ข๐—ฅ)

Separating these into independent offices creates distortion:

Role Reduction
Pastor Limited to preaching
Elder Reduced to board governance
Bishop Elevated as superior authority

Consequences:

โ€ข Artificial hierarchy
โ€ข Clerical elitism
โ€ข Power centralization
โ€ข Leadership abuse
โ€ข Fragmented accountability
โ€ข Doctrinal confusion

๐Ÿ‘‰ This model is foreign to New Testament ecclesiology

๐—ฉ๐—œ. ๐—ช๐—›๐—ฌ ๐—–๐—ข๐—ก๐—™๐—จ๐—ฆ๐—œ๐—ข๐—ก ๐—˜๐—ซ๐—œ๐—ฆ๐—ง๐—ฆ (๐—›๐—œ๐—ฆ๐—ง๐—ข๐—ฅ๐—œ๐—–๐—”๐—Ÿ ๐——๐—˜๐—ฉ๐—˜๐—Ÿ๐—ข๐—ฃ๐— ๐—˜๐—ก๐—ง)

Post-apostolic history gradually institutionalized distinctions:

โ€ข Episcopal system โ€” bishops over pastors
โ€ข Presbyterian system โ€” elder rule emphasized
โ€ข Congregational system โ€” authority rests in members

๐Ÿ‘‰ These are historical polity developments, not direct apostolic commands

Thus:

๐Ÿ‘‰ Many modern distinctions are traditional, not textual

๐—ฉ๐—œ๐—œ. ๐—•๐—œ๐—•๐—Ÿ๐—œ๐—–๐—”๐—Ÿ ๐— ๐—ข๐——๐—˜๐—Ÿ ๐—ฉ๐—ฆ ๐—ฃ๐—ฅ๐—”๐—–๐—ง๐—œ๐—–๐—”๐—Ÿ ๐— ๐—ข๐——๐—˜๐—Ÿ
Biblical Model

โ€ข One office
โ€ข Three integrated functions
โ€ข Plural leadership
โ€ข Christ as the only Head

Practical Model (When Healthy)

โ€ข Multiple leaders
โ€ข Distributed responsibilities
โ€ข Shared authority under Christ

๐Ÿ‘‰ Valid as long as office unity is preserved

๐—ฉ๐—œ๐—œ๐—œ. ๐—•๐—˜๐—ฆ๐—ง ๐—š๐—ข๐—ฉ๐—˜๐—ฅ๐—ก๐—”๐—ก๐—–๐—˜ ๐—ฃ๐—ฅ๐—œ๐—ก๐—–๐—œ๐—ฃ๐—Ÿ๐—˜

๐Ÿ‘‰ Do NOT divide the office
๐Ÿ‘‰ You MAY distribute the function

This preserves:

โ€ข Accountability
โ€ข Balance
โ€ข Biblical fidelity

๐—œ๐—ซ. ๐—˜๐—–๐—–๐—Ÿ๐—˜๐—ฆ๐—œ๐—ข๐—Ÿ๐—ข๐—š๐—œ๐—–๐—”๐—Ÿ ๐—œ๐— ๐—ฃ๐—Ÿ๐—œ๐—–๐—”๐—ง๐—œ๐—ข๐—ก๐—ฆ

A biblical leader must embody all three dimensions:

Pastor (Shepherd)

โ€ข Feeds the Word (Acts 6:4)
โ€ข Cares for souls (Hebrews 13:17)

Elder (Mature Leader)

โ€ข Models godly life (1 Timothy 3:1โ€“7)
โ€ข Leads with wisdom

Bishop (Overseer)

โ€ข Guards doctrine (Titus 1:9)
โ€ข Governs faithfully
Leadership Imbalance Warning

Missing Dimension | Result

Pastor absent | Cold, administrative church
Bishop absent | Disorder, lack of direction
Elder absent | Immature leadership

๐Ÿ‘‰ Healthy leadership requires all three working together

๐—ซ. ๐—”๐—ก๐—”๐—Ÿ๐—ข๐—š๐—œ๐—–๐—”๐—Ÿ ๐—œ๐—Ÿ๐—Ÿ๐—จ๐—ฆ๐—ง๐—ฅ๐—”๐—ง๐—œ๐—ข๐—ก

Consider a physician:

A doctor may:

โ€ข Diagnose
โ€ข Prescribe
โ€ข Perform procedures

๐Ÿ‘‰ Different functions
๐Ÿ‘‰ One identity

Likewise:

Church leaders may serve in varied ways, but remain:

๐Ÿ‘‰ One office under Christ

๐—ซ๐—œ. ๐—–๐—ข๐—ฅ๐—˜ ๐——๐—ข๐—–๐—ง๐—ฅ๐—œ๐—ก๐—”๐—Ÿ ๐—œ๐—ก๐—ฆ๐—œ๐—š๐—›๐—ง

The Bible does NOT teach:

โŒ Three separate offices with ascending authority

But rather:

โœ… One office expressed through three inseparable responsibilities

โœ” Functions may be distributed
โœ– Office must remain unified

A true biblical leader is:

โ€ข Elder in identity
โ€ข Pastor in function
โ€ข Bishop in responsibility

๐Ÿ‘‰ These are not ranks
๐Ÿ‘‰ These are dimensions of one calling

๐—ซ๐—œ๐—œ๐—œ. ๐—–๐—ข๐—ฅ๐—˜ ๐—š๐—ข๐—ฉ๐—˜๐—ฅ๐—ก๐—”๐—ก๐—–๐—˜ ๐—ช๐—”๐—ฅ๐—ก๐—œ๐—ก๐—š

Improper separation leads to:

โ€ข Hierarchical distortion
โ€ข Abuse of authority
โ€ข Loss of plurality
โ€ข Weak accountability
โ€ข Departure from apostolic pattern

๐—ซ๐—œ๐—ฉ. ๐—–๐—›๐—ฅ๐—œ๐—ฆ๐—ง๐—ข๐—Ÿ๐—ข๐—š๐—œ๐—–๐—”๐—Ÿ ๐—–๐—˜๐—ก๐—ง๐—˜๐—ฅ

All leadership finds its unity in Christ:

๐Ÿ‘‰ Colossians 1:18
โ€œAnd he is the head of the body, the churchโ€ฆโ€

๐Ÿ‘‰ Christ is the Chief Shepherd (1 Peter 5:4)
๐Ÿ‘‰ All leaders serve under Him, not above one another

๐Ÿ‘‰ โ€œAre we structuring leadership according to Scriptureโ€”or according to inherited tradition?โ€

๐—ซ๐—ฉ๐—œ. ๐—™๐—œ๐—ก๐—”๐—Ÿ ๐—–๐—ข๐—ก๐—–๐—Ÿ๐—จ๐—ฆ๐—œ๐—ข๐—ก: ๐—ง๐—œ๐—ง๐—Ÿ๐—˜๐—ฆ ๐—ข๐—ฅ ๐— ๐—œ๐—ก๐—œ๐—ฆ๐—ง๐—ฅ๐—ฌ? ๐—” ๐—•๐—œ๐—•๐—Ÿ๐—œ๐—–๐—”๐—Ÿ ๐—–๐—ข๐—ฅ๐—ฅ๐—˜๐—–๐—ง๐—œ๐—ข๐—ก

This study has intentionally focused on pastor, elder, and bishop as they relate to church governance and leadership structure.
It must be clearly stated:
๐Ÿ‘‰ This is not an exhaustive treatment of all ministries
๐Ÿ‘‰ Offices such as prophet, priest, and apostle are not included here
๐Ÿ‘‰ A separate study is necessary to properly address those roles within their own biblical and theological framework

๐—”. ๐— ๐—œ๐—ก๐—œ๐—ฆ๐—ง๐—ฅ๐—ฌ, ๐—ก๐—ข๐—ง ๐—˜๐—ก๐—ง๐—œ๐—ง๐—Ÿ๐—˜๐— ๐—˜๐—ก๐—ง
The central conclusion is this:
๐Ÿ‘‰ These terms in Scripture are functional descriptions, not status titles
In the New Testament:
โ€ข Leadership is defined by service, not position
โ€ข Calling is expressed through responsibility, not recognition
However, in much of modern Christianity:
โ€ข Titles have become identity markers
โ€ข Positions have become forms of recognition
โ€ข Ministry is sometimes confused with entitlement
This creates a dangerous shift:
๐Ÿ‘‰ From servanthood โ†’ status
๐Ÿ‘‰ From calling โ†’ branding
๐Ÿ‘‰ From function โ†’ title

๐—•. ๐—ง๐—›๐—˜ ๐—•๐—œ๐—•๐—Ÿ๐—œ๐—–๐—”๐—Ÿ ๐—ฆ๐—œ๐—Ÿ๐—˜๐—ก๐—–๐—˜ ๐—ข๐—ก ๐—ง๐—œ๐—ง๐—จ๐—Ÿ๐—”๐—ฅ ๐—˜๐— ๐—ฃ๐—›๐—”๐—ฆ๐—œ๐—ฆ
It is important to observe:
๐Ÿ‘‰ The Bible does not command believers to institutionalize these roles as formal titles
๐Ÿ‘‰ Even the fivefold ministry (Ephesians 4:11) is presented as functions given for equipping, not as honorific labels
The emphasis is always:
โ€ข What the person does
โ€ข Not what the person is called by others

๐—–. ๐—ง๐—›๐—˜ ๐—ง๐—ฅ๐—จ๐—˜ ๐—œ๐——๐—˜๐—ก๐—ง๐—œ๐—ง๐—ฌ ๐—ข๐—™ ๐—” ๐— ๐—œ๐—ก๐—œ๐—ฆ๐—ง๐—˜๐—ฅ (๐—š๐—ฅ๐—˜๐—˜๐—ž ๐—œ๐—ก๐—ฆ๐—œ๐—š๐—›๐—ง)
The New Testament consistently brings us back to one foundational identity:
๐Ÿ‘‰ Servant
Key Greek terms:
โ€ข ฮดฮนฮฌฮบฮฟฮฝฮฟฯ‚ (diakonos) โ€” minister, servant
โ†’ One who serves actively, often in practical or spiritual work

โ€ข ฮดฮฟแฟฆฮปฮฟฯ‚ (doulos) โ€” bondservant, slave
โ†’ One who is fully submitted to a master, with no claim of personal status
๐Ÿ‘‰ The root idea of โ€œministerโ€ is not authority, but service

๐——. ๐—ง๐—›๐—˜ ๐—จ๐—Ÿ๐—ง๐—œ๐— ๐—”๐—ง๐—˜ ๐—ง๐—œ๐—ง๐—Ÿ๐—˜ ๐—š๐—ข๐—— ๐—ฅ๐—˜๐—–๐—ข๐—š๐—ก๐—œ๐—ญ๐—˜๐—ฆ
At the end of all ministry, the most important words are not:
โŒ โ€œWell done, good pastorโ€
โŒ โ€œWell done, apostleโ€
โŒ โ€œWell done, bishopโ€
But:
๐Ÿ‘‰ โ€œWell done, thou good and faithful servantโ€ฆโ€ (Matthew 25:21, KJV)
This reveals the highest identity in the Kingdom:
๐Ÿ‘‰ Not title
๐Ÿ‘‰ Not office
๐Ÿ‘‰ Not recognition
But:
๐Ÿ‘‰ Faithful Servanthood

๐—˜. ๐— ๐—œ๐—ก๐—œ๐—ฆ๐—ง๐—ฅ๐—ฌ ๐—™๐—ข๐—ฅ ๐—š๐—ข๐——โ€™๐—ฆ ๐—š๐—Ÿ๐—ข๐—ฅ๐—ฌ (๐—ง๐—˜๐—Ÿ๐—˜๐—ข๐—Ÿ๐—ข๐—š๐—œ๐—–๐—”๐—Ÿ ๐—ฃ๐—จ๐—ฅ๐—ฃ๐—ข๐—ฆ๐—˜)
The ultimate purpose of all ministry is not self-identification, but God-glorification.
โ€ข We serve not to be known
โ€ข We lead not to be elevated
โ€ข We minister not to be titled

But:
To glorify God in every function entrusted to us

As Scripture affirms:
โ€œThat God in all things may be glorifiedโ€ฆโ€ (1 Peter 4:11)

๐—™๐—œ๐—ก๐—”๐—Ÿ ๐——๐—˜๐—–๐—Ÿ๐—”๐—ฅ๐—”๐—ง๐—œ๐—ข๐—ก
Pastor, Elder, Bishop โ€”
Not titles to claim,

But functions to fulfill.
Apostle, Prophet, Evangelist, Teacher โ€”
Not ranks to elevate,
But ministries to operate.

And above all:
We are not called to be titledโ€”
We are called to be faithful servants

๐—–๐—Ÿ๐—ข๐—ฆ๐—œ๐—ก๐—š ๐—˜๐—ซ๐—›๐—ข๐—ฅ๐—ง๐—”๐—ง๐—œ๐—ข๐—ก
Let every leader ask:
โ€œAm I pursuing a titleโ€”or fulfilling a calling?โ€
โ€œAm I building a nameโ€”or serving the Name above all names?โ€
Because in the end:

The greatest honor in the Kingdom is not what we are calledโ€”
๐Ÿ‘‰ But how faithfully we have served.

- Prompted by: Rhoderick Alcaide De Vera

๐—ฆ๐—ข๐—Ÿ๐—œ ๐——๐—˜๐—ข ๐—š๐—Ÿ๐—ข๐—ฅ๐—œ๐—”

21/03/2026

๐Œ๐๐€ ๐‚๐š๐ฅ๐ฅ ๐‚๐ž๐ง๐ญ๐ž๐ซ

แ€™แ€ผแ€”แ€บแ€™แ€ฌแ€กแ€™แ€ปแ€ญแ€ฏแ€ธแ€žแ€ฌแ€ธแ€œแ€ฑแ€€แ€ผแ€ฑแ€ฌแ€„แ€บแ€ธแ€€แ€ญแ€ฏ แ€กแ€…แ€‰แ€บแ€™แ€•แ€ผแ€แ€บแ€กแ€ฌแ€ธแ€•แ€ฑแ€ธแ€…แ€ฎแ€ธแ€”แ€„แ€บแ€ธแ€œแ€ญแ€ฏแ€€แ€บแ€•แ€ซแ€€แ€ผแ€แ€ฒแ€ท แ€แ€›แ€ฎแ€ธแ€žแ€ฝแ€ฌแ€ธแ€•แ€ผแ€Šแ€บแ€žแ€ฐแ€™แ€ปแ€ฌแ€ธ แ€แ€›แ€ฎแ€ธแ€…แ€‰แ€บแ€กแ€แ€ปแ€€แ€บ แ€กแ€œแ€€แ€บแ€™แ€ปแ€ฌแ€ธ แ€…แ€ฏแ€ถแ€…แ€™แ€บแ€ธแ€™แ€ฑแ€ธแ€™แ€ผแ€”แ€บแ€ธแ€›แ€”แ€บ แ€™แ€ผแ€”แ€บแ€™แ€ฌแ€กแ€™แ€ปแ€ญแ€ฏแ€ธแ€žแ€ฌแ€ธแ€œแ€ฑแ€€แ€ผแ€ฑแ€ฌแ€„แ€บแ€ธ Call Center แ€–แ€ฏแ€”แ€บแ€ธแ€”แ€ถแ€•แ€ซแ€แ€บแ€™แ€ปแ€ฌแ€ธแ€€แ€ญแ€ฏ แ€แ€”แ€„แ€บแ€นแ€œแ€ฌแ€”แ€ฑแ€ท แ€™แ€พ แ€žแ€ฑแ€ฌแ€€แ€ผแ€ฌแ€”แ€ฑแ€ท (แ€แ‰:แƒแ€ - แแ†:แƒแ€) แ€”แ€พแ€„แ€ทแ€บ แ€…แ€”แ€ฑแ€”แ€ฑแ€ท แ€™แ€พ แ€แ€”แ€„แ€บแ€นแ€‚แ€”แ€ฝแ€ฑแ€”แ€ฑแ€ท (แ€แ‰:แƒแ€ - แแƒ:แ€แ€) แ€กแ€แ€ฝแ€„แ€บแ€ธแ€†แ€€แ€บแ€žแ€ฝแ€šแ€บแ€”แ€ญแ€ฏแ€„แ€บแ€€แ€ผแ€ฑแ€ฌแ€„แ€บแ€ธ แ€กแ€žแ€ญแ€•แ€ฑแ€ธแ€กแ€•แ€บแ€•แ€ซแ€แ€šแ€บแ€›แ€พแ€„แ€บแ‹

[English]

Myanmar National Airlinesโ€™ Call Center is operating Monday to Friday from 9:30am to 4:30pm and Saturday to Sunday from 9:30am to 1:00pm for your travel inquiries.

โ˜Ž๏ธMNA Call Center:
+951 378603 โ€“ 04 โ€“ 06 โ€“ 07 โ€“ 08 | +951 377840 โ€“ 41 โ€“ 42

02/11/2025

Welcome to GNI Foundation

K*m 2000 val apha Gesemani huan Olive kungpi cizen veh aw ... sauvei ahin dan
28/10/2025

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Garden of Gethsemane, Mount of Olives Jerusalem

29/09/2025

แ€™แ€ผแ€ฑแ€งแ€€ แ€แ€ฝแ€€แ€บแ€”แ€Šแ€บแ€ธ
============
แ€™แ€ผแ€ฑแ€งแ€€ แ€˜แ€šแ€บแ€œแ€ญแ€ฏแ€แ€ฝแ€€แ€บแ€›แ€žแ€œแ€ฒ แ€แ€แ€แ€ แ€™แ€ฑแ€ธแ€œแ€ฌแ€œแ€ญแ€ฏแ€ท แ€กแ€แ€แ€บแ€”แ€ญแ€ฏแ€„แ€บแ€†แ€ฏแ€ถแ€ธ แ€›แ€พแ€„แ€บแ€ธแ€œแ€„แ€บแ€ธแ€กแ€ฑแ€ฌแ€„แ€บ แ€แ€ฝแ€€แ€บแ€”แ€Šแ€บแ€ธแ€€แ€ญแ€ฏ แ€–แ€ฑแ€ฌแ€บแ€•แ€ผแ€œแ€ญแ€ฏแ€€แ€บแ€•แ€ซแ€แ€šแ€บแ‹ แ€™แ€ญแ€™แ€ญแ€™แ€ผแ€ฑแ€€แ€ฝแ€€แ€บแ€›แ€ฒแ€ท แ€•แ€ฏแ€ถแ€žแ€แ€นแ€แ€”แ€บแ€€แ€ญแ€ฏแ€€แ€ผแ€Šแ€บแ€ทแ€•แ€ผแ€ฎแ€ธ แ€งแ€€ แ€แ€ฝแ€€แ€บแ€”แ€ญแ€ฏแ€„แ€บแ€•แ€ซแ€แ€šแ€บแ‹

แแ‹ แ€œแ€ฑแ€ธแ€”แ€ฌแ€ธแ€Šแ€ฎ แ€…แ€แ€ฏแ€›แ€”แ€บแ€ธแ€•แ€ฏแ€ถแ€…แ€ถ แ€–แ€ผแ€…แ€บแ€•แ€ซแ€€
แ‚แ€แˆ.แ‡แ' แ‚แ€แˆ.แ‡แ' = แ„แƒแ…แ…แ‰.แˆแ† = แ แ€งแ€€
แ€ฅแ€•แ€™แ€ฌ ~~~~
แแ€แ€ แ€•แ€ฑ แ€•แ€แ€บแ€œแ€Šแ€บแ€กแ€€แ€ฝแ€€แ€บ แ€–แ€ผแ€…แ€บแ€œแ€ปแ€„แ€บ
แแ€แ€' แแ€แ€' แ„แƒแ…แ†แ€=แ€. แ‚แ‚แ‰ แ€งแ€€

แ‚แ‹ แ€”แ€พแ€…แ€บแ€”แ€ฌแ€ธแ€Šแ€ฎ แ€…แ€แ€ฏแ€‚แ€ถแ€•แ€ฏแ€ถแ€…แ€ถ แ€–แ€ผแ€…แ€บแ€œแ€ปแ€พแ€„แ€บ
(แ€œ x แ€”)' แ„แƒแ…แ†แ€ = __แ€งแ€€
แ€ฅแ€•แ€™แ€ฌ ~~~~
(แ„แ€'แ†แ€')แ„แƒแ…แ†แ€ = แ€.แ€แ…แ… แ€งแ€€
(แ†แ€'แˆแ€')แ„แƒแ…แ†แ€ = แ€.แแ แ€งแ€€
(แ†แ€'แแ€แ€')แ„แƒแ…แ†แ€ = แ€.แแƒแ‡ แ€งแ€€
(แˆแ€'แแ€แ€')แ„แƒแ…แ†แ€ = แ€.แแˆแƒแ† แ€งแ€€
(แแ€แ€'แแ€แ€')แ„แƒแ…แ†แ€ = แ€. แ‚แ‚แ‰ แ€งแ€€

แƒแ‹ แ€”แ€พแ€…แ€บแ€”แ€ฌแ€ธแ€•แ€ผแ€ญแ€ฏแ€„แ€บ แ€แ€ผแ€ฌแ€—แ€ฎแ€‡แ€ฎแ€šแ€ถแ€•แ€ฏแ€ถแ€…แ€ถ แ€–แ€ผแ€…แ€บแ€•แ€ซแ€€
แ€•แ€ผแ€ญแ€ฏแ€„แ€บแ€”แ€ฌแ€ธแ€”แ€พแ€…แ€บแ€แ€ฏแ€•แ€ฑแ€ซแ€„แ€บแ€ธ แ‚ x แ€”แ€ถ แ„แƒแ…แ†แ€ = แ€งแ€€
แ€ฅแ€•แ€™แ€ฌ ~~~~
[{(แ…แ€+แˆแ€)แ‚}แ†แ€]แ„แƒแ…แ†แ€ = แ€.แ€แˆแ‰ แ€งแ€€

แ„แ‹ แ€แ€ผแ€ญแ€‚แ€ถแ€•แ€ฏแ€ถแ€…แ€ถแ€–แ€ผแ€…แ€บแ€œแ€ปแ€„แ€บแ… x แ€กแ€แ€ผแ€ฑ x แ€กแ€™แ€ผแ€„แ€บแ€ท แ„แƒแ…แ†แ€ = แ€งแ€€
แ€กแ€แ€ผแ€ฑ x แ€กแ€™แ€ผแ€„แ€บแ€ท แ„แƒแ…แ†แ€ = แ€งแ€€
แ€ฅแ€•แ€™แ€ฌ ~~~~แ… แแ€แ€' แ…แ€' แ„แƒแ…แ†แ€ = แ€.แ€แ…แ‡ แ€งแ€€

แ…แ‹ แ€‘แ€ฑแ€ฌแ€„แ€บแ€ทแ€™แ€พแ€”แ€บ แ€แ€ผแ€ญแ€‚แ€ถแ€•แ€ฏแ€ถแ€…แ€ถแ€–แ€ผแ€…แ€บแ€œแ€ปแ€พแ€„แ€บ
แ€กแ€แ€ผแ€ฑ x แ€กแ€™แ€ผแ€„แ€บแ€ท = แ€›แ€œแ€ฌแ€’แ€บ แ‚ แ„แƒแ…แ†แ€
แ€ฅแ€•แ€™แ€ฌ ~~~~
แแ€แ€' แ…แ€' = แ…แ€แ€แ€'
แ…แ€แ€แ€ แ‚ แ„แƒแ…แ†แ€ = แ€.แ€แ…แ‡ แ€งแ€€

แ†แ‹ แ€…แ€€แ€บแ€แ€ญแ€ฏแ€„แ€บแ€ธแ€•แ€ฏแ€ถแ€…แ€ถแ€–แ€ผแ€…แ€บแ€œแ€ปแ€พแ€„แ€บ
แ€กแ€แ€ปแ€„แ€บแ€ธแ€แ€€แ€บ x แ€กแ€แ€ปแ€„แ€บแ€ธแ€แ€€แ€บ x แ€•แ€ญแ€ฏแ€„แ€บแ€€แ€ญแ€”แ€บแ€ธ แ„แƒแ…แ†แ€ = แ€งแ€€ แ€–แ€ผแ€…แ€บแ€žแ€Šแ€บแ‹ แ€‘แ€ญแ€ฏแ€ทแ€€แ€ผแ€ฑแ€ฌแ€„แ€บแ€ท
แ€กแ€แ€ปแ€„แ€บแ€ธ x แ€กแ€แ€ปแ€„แ€บแ€ธ x .แ‡แˆแ…แ„ แ„แƒแ…แ†แ€ = แ€งแ€€ (แ€•แ€ฏแ€ถแ€žแ€ฑแ€™แ€ผแ€พแ€ฑแ€ฌแ€€แ€บแ€€แ€ญแ€”แ€บแ€ธ .แ‡แˆแ…แ„)
(.แ‡แˆแ…แ„ แ€žแ€Šแ€บ แ€•แ€ญแ€ฏแ€„แ€บแ€€แ€ญแ€”แ€บแ€ธ แƒ.แแ„แแ† แ€€แ€ญแ€ฏ แ€กแ€”แ€ฌแ€ธแ€œแ€ฑแ€ธแ€–แ€€แ€บ แ„ แ€”แ€พแ€„แ€บแ€ทแ€…แ€ฌแ€ธแ€•แ€ผแ€ฎแ€ธ แ€…แ€แ€ฏแ€›แ€”แ€บแ€ธแ€€แ€ฝแ€€แ€บแ€–แ€ฑแ€ซแ€บแ€แ€ผแ€„แ€บแ€ธแ€–แ€ผแ€…แ€บแ€žแ€Šแ€บ)
แ€ฅแ€•แ€™แ€ฌ ~~~~
แ…แ€' แ…แ€' .แ‡แˆแ…แ„ แ„แƒแ…แ†แ€ = แ€.แ€แ„แ… แ€งแ€€

แ‡แ‹ แ€กแ€”แ€ฌแ€ธ แ… แ€™แ€ปแ€€แ€บแ€”แ€พแ€ฌ แ† แ€™แ€ปแ€€แ€บแ€”แ€พแ€ฌแ€”แ€พแ€„แ€บแ€ท แ€กแ€”แ€ฌแ€ธแ€™แ€ปแ€ฌแ€ธแ€…แ€ฝแ€ฌแ€•แ€ซแ€žแ€ฑแ€ฌ แ€•แ€ฏแ€ถแ€…แ€ถแ€–แ€ผแ€…แ€บแ€œแ€ปแ€พแ€„แ€บ แ€…แ€แ€ฏแ€‚แ€ถ แ€…แ€แ€ฏแ€›แ€”แ€บแ€ธ แ€แ€ผแ€ญแ€‚แ€ถแ€…แ€žแ€Šแ€บ แ€กแ€€แ€ฝแ€€แ€บแ€…แ€ญแ€•แ€บแ€•แ€ผแ€ฎแ€ธแ€แ€ฝแ€€แ€บแŠ แ€›แ€œแ€ฌแ€’แ€บแ€กแ€ฌแ€ธแ€œแ€ฏแ€ถแ€ธแ€€แ€ญแ€ฏแ€•แ€ฑแ€ซแ€„แ€บแ€ธแ แ„แƒแ…แ†แ€ แ€–แ€ผแ€„แ€บแ€ทแ€…แ€ฌแ€ธแ‹

แ แ€งแ€€ = แ„แƒแ…แ†แ€ แ€…แ€แ€ฏแ€›แ€”แ€บแ€ธแ€•แ€ฑ แ€–แ€ผแ€…แ€บแ€•แ€ซแ€žแ€Šแ€บแ‹

แ€กแ€Šแ€ฝแ€พแ€”แ€บแ€ธ....
แ€œ = แ€กแ€œแ€ปแ€ฌแ€ธ
แ€”= แ€กแ€”แ€ถ
' = แ€•แ€ฑ
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