<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font> holds <br/> ( <font color="Olive" title="b1">T</font> is  <a href="cat_1.html#V9" title="CAT_1:attr.9">full</a>  iff  for <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">c9</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds   <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Olive" title="b1">T</font>,<font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">c9</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b3">c9</font></span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">T</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">T</font> is  <a href="cat_1.html#V9" title="CAT_1:attr.9">full</a>  iff  for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">c9</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds   <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">c9</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">c9</font></span>)</span>) )</span><br/>





<span class="kw">thus </span><a NAME="E1:102"/>
( <font color="Maroon" title="c3">T</font> is  <a href="cat_1.html#V9" title="CAT_1:attr.9">full</a>  implies  for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">c9</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds   <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">c9</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">c9</font></span>)</span>) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">c9</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds   <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">c9</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">c9</font></span>)</span>) ) implies <font color="Maroon" title="c3">T</font> is  <a href="cat_1.html#V9" title="CAT_1:attr.9">full</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:102_1"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<font color="Maroon" title="c3">T</font> is  <a href="cat_1.html#V9" title="CAT_1:attr.9">full</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">c9</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds   <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">c9</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">c9</font></span>)</span>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">c9</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>)</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:102_1_1"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>)</span><br/><a NAME="E2:102_1_1"/>
 for <font color="Olive" title="b1">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span> iff <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span> iff <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>) )</span><br/>

<span class="kw">thus </span><a NAME="E1:102_1_1_1"/>
( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span> implies <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>) implies <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:102_1_1_1_1"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:102_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:102_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:102_1_1_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E3:102_1_1_1_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:102_1_1_1_1"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E4:102_1_1_1_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:102_1_1_1"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span></span><br/>

<a NAME="E3:102_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">g</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D4" target="_self" title="CAT_1:def.4">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font><span class="kw"> such that </span><br/><a NAME="E5:102_1_1_1"/><span class="lab"><font color="Green" title="E61">A5</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E1:102_1_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="cat_1.html#D20" target="_self" title="CAT_1:def.20">Def23</a></span>;<br/>
<a NAME="E6:102_1_1_1"/><span class="lab"><font color="Green" title="E62">A6</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E1:102_1_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="cat_1.html#D20" target="_self" title="CAT_1:def.20">Def23</a></span>;<br/>
<a NAME="E7:102_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D4" target="_self" title="CAT_1:def.4">Def7</a></span>;<br/>
<a NAME="E8:102_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:102_1_1_1"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:102_1_1_1"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E6:102_1_1_1"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="ref" href="cat_1.html#T88" target="_self" title="CAT_1:th.88">Th131</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:102_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:102_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:102"/><span class="lab"><font color="Green" title="E59">A7</font></span>: 
 for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">c9</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds   <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">c9</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b2">c9</font></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">T</font> is  <a href="cat_1.html#V9" title="CAT_1:attr.9">full</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">c9</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D20" target="_self" title="CAT_1:def.20">CAT_1:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  for <font color="Olive" title="b1">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font> holds <br/> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <font color="Olive" title="b2">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font> st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:102"/><span class="lab"><font color="Green" title="E60">A8</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font> holds <br/> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <font color="Olive" title="b2">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font> st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <font color="Olive" title="b1">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font> st <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> )</span><br/>

<a NAME="E4:102"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c5">c9</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="ref" href="cat_1.html#D4" target="_self" title="CAT_1:def.4">Def7</a></span>;<br/>
<a NAME="E5:102"/><span class="kw">then </span>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:102"><span class="lab"><font color="Green" title="E59">A7</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:102"/><span class="lab"><font color="Green" title="E61">A9</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:102"/><span class="lab"><font color="Green" title="E62">A10</font></span>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">hom</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:102"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:102"><span class="lab"><font color="Green" title="E61">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font> st <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span><br/>

<a NAME="E10:102"/><span class="lab"><font color="Green" title="E63">A11</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="txt" href="#E7:102"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">f</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">c9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D4" target="_self" title="CAT_1:def.4">Def7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font></span><br/>

<span class="kw">thus </span><a NAME="E12:102"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:102"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E10:102"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="ref" href="cat_1.html#T88" target="_self" title="CAT_1:th.88">Th131</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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