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

<b>set </b><font color="Maroon" title="c5">DD</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font>;<br/>
<b>set </b><font color="Maroon" title="c6">CC</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font>;<br/>
<b>set </b><font color="Maroon" title="c7">Cm</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font>;<br/>
<b>set </b><font color="Maroon" title="c8">I</font> = the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">O</font> = the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="cat_5.html#E4:13"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">o2</font> = <font color="Maroon" title="c4">o1</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C2</font> <b>by </b><i><a class="txt" href="cat_5.html#E4:13"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">o</font> = <font color="Maroon" title="c4">o1</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font> <b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="cat_2.html#T12" title="CAT_2:th.12">CAT_2:12</a></i>;<br/>
<a NAME="E4:13_1_1"/>
(  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c11">o</font> &amp;  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c10">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c11">o</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">M</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:13_1_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C1</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C1</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:13_1_1"/><b>then </b><i><font color="Green" title="E8">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">M</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">M</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">O</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E8:13_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1_1"/><i><font color="Green" title="E9">A4</font></i>: 
( <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">m1</font> = <font color="Maroon" title="c14">m</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_1"><i><font color="Green" title="E9">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">m2</font> = <font color="Maroon" title="c14">m</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C2</font> <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">m</font> = <font color="Maroon" title="c15">m1</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font> <b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="cat_2.html#T14" title="CAT_2:th.14">CAT_2:14</a></i>;<br/>
<a NAME="E7:13_1_1_1"/>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c15">m1</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c15">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c17">m</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c16">m2</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="cat_2.html#T15" title="CAT_2:th.15">CAT_2:15</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E8:13_1_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">O</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">DD</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c9">O</font> <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E10:13_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1_2"/><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">m1</font> = <font color="Maroon" title="c15">m</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_2"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">m2</font> = <font color="Maroon" title="c15">m</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C2</font> <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_2"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">m</font> = <font color="Maroon" title="c16">m1</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font> <b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="cat_2.html#T14" title="CAT_2:th.14">CAT_2:14</a></i>;<br/>
<a NAME="E7:13_1_1_2"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c16">m1</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c18">m</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c17">m2</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_2"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="cat_2.html#T15" title="CAT_2:th.15">CAT_2:15</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E8:13_1_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">O</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">CC</font> = the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c9">O</font> <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E12:13_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1_3"/><i><font color="Green" title="E9">A6</font></i>: 
( <font color="Maroon" title="c16">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C2</font></span>)</span> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">m1</font> = <font color="Maroon" title="c16">m</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_3"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">m2</font> = <font color="Maroon" title="c16">m</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C2</font> <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">m</font> = <font color="Maroon" title="c17">m1</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C</font> <b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="cat_2.html#T12" title="CAT_2:th.12">CAT_2:12</a></i>;<br/>
<a NAME="E7:13_1_1_3"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c17">m1</font> &amp;  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c17">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c19">m</font> &amp;  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c18">m2</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E8:13_1_1_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c15">I</font> = the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font> <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E14:13_1_1"/><i><font color="Green" title="E9">A7</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E15:13_1_1"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C1</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E16:13_1_1"/><b>then </b><i><font color="Green" title="E10">A8</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c12">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></i>;<br/>
<a NAME="E17:13_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_4"/>
<font color="Maroon" title="c16">x</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">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1_4"/><i><font color="Green" title="E11">A9</font></i>: 
( <font color="Maroon" title="c17">m</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">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:13_1_1_4"/><i><font color="Green" title="E12">A10</font></i>: 
( <font color="Maroon" title="c17">m</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font> &amp; <font color="Maroon" title="c17">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font> &amp; <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E16:13_1_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_4"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">m1</font> = <font color="Maroon" title="c17">m</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c19">m2</font> = <font color="Maroon" title="c17">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">n1</font> = <font color="Maroon" title="c17">m</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c21">n2</font> = <font color="Maroon" title="c17">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C2</font> <b>by </b><i><a class="txt" href="cat_5.html#E4:13_1_1_4"><i><font color="Green" title="E12">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">mm</font> = <font color="Maroon" title="c18">m1</font>, <font color="Maroon" title="c23">n</font> = <font color="Maroon" title="c19">m2</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C</font> <b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="cat_2.html#T14" title="CAT_2:th.14">CAT_2:14</a></i>;<br/>
<a NAME="E8:13_1_1_4"/><i><font color="Green" title="E13">A11</font></i>: 
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E14:13_1_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_4"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:13_1_1_4"/><b>then </b><i><font color="Green" title="E14">A12</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c18">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c19">m2</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_4"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="cat_5.html#E4:13_1_1_4"><i><font color="Green" title="E12">A10</font></i></a>, <a class="ref" href="cat_1.html#T40" title="CAT_1:th.40">CAT_1:40</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E10:13_1_1_4"/><i><font color="Green" title="E15">A13</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c18">m1</font>,<font color="Maroon" title="c19">m2</font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_4"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="cat_5.html#E4:13_1_1_4"><i><font color="Green" title="E12">A10</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E11:13_1_1_4"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c18">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c22">mm</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c20">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c22">mm</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c19">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c23">n</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c21">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c23">n</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="cat_2.html#T15" title="CAT_2:th.15">CAT_2:15</a></i>;<br/>
<a NAME="E12:13_1_1_4"/><b>then </b>
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">m1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c19">m2</font> &amp; <font color="Maroon" title="c18">m1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c19">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">mm</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c22">mm</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">n1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c21">n2</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="cat_5.html#E4:13_1_1_4"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="cat_5.html#E8:13_1_1_4"><i><font color="Green" title="E13">A11</font></i></a>, <a class="txt" href="cat_5.html#E9:13_1_1_4"><i><font color="Green" title="E14">A12</font></i></a>, <a class="txt" href="cat_5.html#E10:13_1_1_4"><i><font color="Green" title="E15">A13</font></i></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a>, <a class="ref" href="cat_2.html#T17" title="CAT_2:th.17">CAT_2:17</a></i>;<br/>
<b>hence </b><a NAME="E13:13_1_1_4"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">M</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c16">Cm</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c12">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c12">M</font> <b>by </b><i><a class="txt" href="cat_5.html#E16:13_1_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">CAT</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #);<br/>
<a NAME="E19:13_1_1"/>
 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) is <a href="cat_1.html#V3" title="CAT_1:attr.3">Category-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_5"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>
<div class="add"><b>let </b><font color="Maroon" title="c18">f</font>, <font color="Maroon" title="c19">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #);<br/><b>reconsider </b><font color="Maroon" title="c20">g'</font> = <font color="Maroon" title="c19">g</font>, <font color="Maroon" title="c21">f'</font> = <font color="Maroon" title="c18">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E2:13_1_1_5_1"/><i><font color="Green" title="E12">A15</font></i>: 
( <font color="Maroon" title="c19">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> &amp; <font color="Maroon" title="c18">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1_1_5_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #)
 ;<br/><a NAME="E2:13_1_1_5_1_1"/><b>then </b><i><font color="Green" title="E13">A16</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E14:13_1_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">g'</font>
<b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5_1"><i><font color="Green" title="E12">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">f'</font>
<b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5_1_1"><i><font color="Green" title="E13">A16</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">f</font>
<b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5_1"><i><font color="Green" title="E12">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E4:13_1_1_5_1"/>
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">f</font>
 ;<br/><a NAME="E5:13_1_1_5_1"/><b>then </b>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">g'</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">f'</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E6:13_1_1_5_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5_1"><i><font color="Green" title="E12">A15</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E7:13_1_1_5_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #)
 <b>by </b><i><a class="txt" href="cat_5.html#E14:13_1_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c18">f</font>, <font color="Maroon" title="c19">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #);<br/><b>reconsider </b><font color="Maroon" title="c20">g'</font> = <font color="Maroon" title="c19">g</font>, <font color="Maroon" title="c21">f'</font> = <font color="Maroon" title="c18">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>assume </b><a NAME="E2:13_1_1_5_2"/><i><font color="Green" title="E13">A18</font></i>: 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">f</font>
 ;<br/><a NAME="E3:13_1_1_5_2"/><b>then </b><i><font color="Green" title="E14">A19</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #)
 <b>by </b><i><a class="txt" href="cat_5.html#E1:13_1_1_5"><i><font color="Green" title="E11">A14</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">h</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <b>by </b><i><a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/><a NAME="E5:13_1_1_5_2"/>
( <font color="Maroon" title="c22">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">g'</font>,<font color="Maroon" title="c21">f'</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">f'</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">g'</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">h</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">f'</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">g'</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">h</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_5_2"><i><font color="Green" title="E14">A19</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E6:13_1_1_5_2"/>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">f</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5_2"><i><font color="Green" title="E13">A18</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c18">f</font>, <font color="Maroon" title="c19">g</font>, <font color="Maroon" title="c20">h</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #);<br/><b>reconsider </b><font color="Maroon" title="c21">h'</font> = <font color="Maroon" title="c20">h</font>, <font color="Maroon" title="c22">g'</font> = <font color="Maroon" title="c19">g</font>, <font color="Maroon" title="c23">f'</font> = <font color="Maroon" title="c18">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>assume </b><a NAME="E2:13_1_1_5_3"/><i><font color="Green" title="E13">A20</font></i>: 
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">f</font> )
 ;<br/><a NAME="E3:13_1_1_5_3"/><b>then </b><i><font color="Green" title="E14">A21</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">h</font>,<font color="Maroon" title="c19">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) )
 <b>by </b><i><a class="txt" href="cat_5.html#E1:13_1_1_5"><i><font color="Green" title="E11">A14</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c24">hg</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">h</font>,<font color="Maroon" title="c19">g</font>, <font color="Maroon" title="c25">gf</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <b>by </b><i><a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">hg'</font> = <font color="Maroon" title="c24">hg</font>, <font color="Maroon" title="c27">gf'</font> = <font color="Maroon" title="c25">gf</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:13_1_1_5_3"/>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">hg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">gf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5"><i><font color="Green" title="E12">A17</font></i></a>, <a class="txt" href="cat_5.html#E2:13_1_1_5_3"><i><font color="Green" title="E13">A20</font></i></a></i>;<br/><a NAME="E7:13_1_1_5_3"/><b>then </b><i><font color="Green" title="E15">A22</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">hg</font>,<font color="Maroon" title="c18">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">h</font>,<font color="Maroon" title="c25">gf</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) )
 <b>by </b><i><a class="txt" href="cat_5.html#E1:13_1_1_5"><i><font color="Green" title="E11">A14</font></i></a>, <a class="txt" href="cat_5.html#E2:13_1_1_5_3"><i><font color="Green" title="E13">A20</font></i></a></i>;<br/><a NAME="E8:13_1_1_5_3"/><i><font color="Green" title="E16">A23</font></i>: 
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">h'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">h</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">f</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>thus </b>the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">h</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font></span>)</span> = 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c21">h'</font>,<font color="Maroon" title="c27">gf'</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1_5_3"><i><font color="Green" title="E15">A22</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">h'</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">g'</font>,<font color="Maroon" title="c23">f'</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_5_3"><i><font color="Green" title="E14">A21</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">h'</font>,<font color="Maroon" title="c22">g'</font></span>)</span>,<font color="Maroon" title="c23">f'</font>
<b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5_3"><i><font color="Green" title="E13">A20</font></i></a>, <a class="txt" href="cat_5.html#E8:13_1_1_5_3"><i><font color="Green" title="E16">A23</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c26">hg'</font>,<font color="Maroon" title="c23">f'</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_5_3"><i><font color="Green" title="E14">A21</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">h</font>,<font color="Maroon" title="c19">g</font></span>)</span>,<font color="Maroon" title="c18">f</font>
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1_5_3"><i><font color="Green" title="E15">A22</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c18">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c19">b'</font> = <font color="Maroon" title="c18">b</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>thus </b>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c18">b</font>
<b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
;<br/>

<b>thus </b>the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c18">b</font>
<b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c20">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #);<br/><b>reconsider </b><font color="Maroon" title="c21">f'</font> = <font color="Maroon" title="c20">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>assume </b><a NAME="E2:13_1_1_5_6"/>
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font>
 ;<br/><a NAME="E3:13_1_1_5_6"/><b>then </b>
( the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">b'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b'</font> &amp; the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font> &amp; <font color="Maroon" title="c20">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> &amp; the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:13_1_1_5_6"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span>,<font color="Maroon" title="c20">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span>,<font color="Maroon" title="c20">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">b'</font></span>)</span>,<font color="Maroon" title="c21">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f'</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:13_1_1_5_6"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span>,<font color="Maroon" title="c20">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span>,<font color="Maroon" title="c20">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="cat_5.html#E14:13_1_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E6:13_1_1_5_6"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span>,<font color="Maroon" title="c20">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c20">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c21">g'</font> = <font color="Maroon" title="c20">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>assume </b><a NAME="E9:13_1_1_5"/>
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font>
 ;<br/>

<a NAME="E10:13_1_1_5"/><b>then </b>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b'</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">b'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b'</font> &amp; the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font> &amp; <font color="Maroon" title="c20">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> &amp; the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:13_1_1_5"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">g</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">g</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">g'</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">b'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g'</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:13_1_1_5"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">g</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">g</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="cat_5.html#E14:13_1_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:13_1_1_5"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">g</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c13">DD</font>,<font color="Maroon" title="c14">CC</font>,<font color="Maroon" title="c16">Cm</font>,<font color="Maroon" title="c15">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:13_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> st <br/>( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C2</font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">C1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">C1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C2</font> &amp; the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">C1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C2</font> )
 ;<br/>


</div>
