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

<span class="kw">let </span><font color="Maroon" title="c1">A</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>, <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>  st  <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Olive" title="b1">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="cat_1.html#V10" title="CAT_1:attr.10">faithful</a>  holds <br/><font color="Olive" title="b1">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>, <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="cat_1.html#V10" title="CAT_1:attr.10">faithful</a>  implies <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:10"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">F</font> is  <a href="cat_1.html#V10" title="CAT_1:attr.10">faithful</a>  or <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:10"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="cat_1.html#V10" title="CAT_1:attr.10">faithful</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>

<a NAME="E3:10"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</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> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">x2</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> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x1</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> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c4">x2</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> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font> implies <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
( <font color="Maroon" title="c3">x1</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> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c4">x2</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> <font color="Maroon" title="c2">F</font> )
 <span class="kw">and </span><br/><a NAME="E2:10_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">m1</font> = <font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c6">m2</font> = <font color="Maroon" title="c4">x2</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E8">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">set </span><font color="Maroon" title="c7">o1</font> =  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">o2</font> =  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">o3</font> =  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">o4</font> =  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">m19</font> = <font color="Maroon" title="c5">m1</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font>, <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T22" title="CAT_1:th.22">CAT_1:22</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">m29</font> = <font color="Maroon" title="c6">m2</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font>, <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T22" title="CAT_1:th.22">CAT_1:22</a></span>;<br/>
<a NAME="E6:10_1"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T19" title="CAT_1:th.19">CAT_1:19</a></span>;<br/>
<a NAME="E7:10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A6</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="c2">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</font></span>)</span></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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></span>;<br/>
<a NAME="E8:10_1"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></span>;<br/>
<a NAME="E9:10_1"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></span>;<br/>
<a NAME="E10:10_1"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T19" title="CAT_1:th.19">CAT_1:19</a></span>;<br/>
<a NAME="E11:10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A10</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="c2">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span></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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T126" title="CAT_1:th.126">CAT_1:126</a></span>;<br/>
<span class="lab"><font color="Green" title="E16">A11</font></span>: <font color="Maroon" title="c2">F</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c11">m19</font> = 
<font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">m2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E6:10_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">m29</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font></span>)</span> = 
<font color="Maroon" title="c2">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></span>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">m29</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:10_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E7:10_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E9:10_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>
;<br/>
<a NAME="E14:10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A12</font></span>: 
( <font color="Maroon" title="c6">m2</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font>, <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="cat_1.html#T22" title="CAT_1:th.22">CAT_1:22</a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></span>;<br/>
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</font></span>)</span> = 
<font color="Maroon" title="c2">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D20" title="CAT_1:def.20">CAT_1:def 20</a></span>
<br/>.= 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c12">m29</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:10_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E7:10_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E8:10_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="cat_1.html#T23" title="CAT_1:th.23">CAT_1:23</a></span>
;<br/>
<a NAME="E16:10_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">m1</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font>, <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</font> &amp; <font color="Maroon" title="c6">m2</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font>, <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E14:10_1"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="cat_1.html#T22" title="CAT_1:th.22">CAT_1:22</a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:10_1"/>
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E6:10_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="cat_1.html#D24" title="CAT_1:def.24">CAT_1:def 24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:10"/>
<font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
