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

<b>let </b><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>;<br/>



<b>set </b><font color="Maroon" title="c4">S1</font> =  <a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font>;<br/>
<b>set </b><font color="Maroon" title="c5">S2</font> =  <a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</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>set </b><font color="Maroon" title="c6">A1</font> =  <a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">A2</font> =  <a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c2">C2</font>;<br/>
<b>set </b><font color="Maroon" title="c8">f</font> =  <a href="catalg_1.html#K11" title="CATALG_1:func.11">Upsilon</a> <font color="Maroon" title="c3">F</font>;<br/>
<b>set </b><font color="Maroon" title="c9">g</font> =  <a href="catalg_1.html#K12" title="CATALG_1:func.12">Psi</a> <font color="Maroon" title="c3">F</font>;<br/>
<b>set </b><font color="Maroon" title="c10">B1</font> = <span class="p1">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c2">C2</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font></span>)</span>,<span class="p1">(<span class="default"><a href="catalg_1.html#K11" title="CATALG_1:func.11">Upsilon</a> <font color="Maroon" title="c3">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="catalg_1.html#K12" title="CATALG_1:func.12">Psi</a> <font color="Maroon" title="c3">F</font></span>)</span>;<br/>
<a NAME="E1:74"/>
 <a href="catalg_1.html#K11" title="CATALG_1:func.11">Upsilon</a> <font color="Maroon" title="c3">F</font>, <a href="catalg_1.html#K12" title="CATALG_1:func.12">Psi</a> <font color="Maroon" title="c3">F</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a>  <a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</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="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C2</font>
 
<b>by </b><i><a class="ref" href="catalg_1.html#T33" target="_self" title="CATALG_1:th.33">Th33</a></i>;<br/>
<a NAME="E2:74"/><b>then </b><i><font color="Green" title="E39">A1</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c2">C2</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K11" title="CATALG_1:func.11">Upsilon</a> <font color="Maroon" title="c3">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K12" title="CATALG_1:func.12">Psi</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c2">C2</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K11" title="CATALG_1:func.11">Upsilon</a> <font color="Maroon" title="c3">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="instalg1.html#D3" title="INSTALG1:def.3">INSTALG1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">H</font> = <font color="Maroon" title="c3">F</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font></span>)</span>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C1</font></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c12">i</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="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E3:74"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c12">i</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C1</font><b> such that </b><br/><a NAME="E6:74"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K9" title="CATALG_1:func.9">homsym</a> <font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T24" target="_self" title="CATALG_1:th.24">Th24</a></i>;<br/>
<a NAME="E7:74"/><i><font color="Green" title="E41">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font></span>)</span>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="catalg_1.html#D1" target="_self" title="CATALG_1:def.1">Def1</a></i>;<br/>
<a NAME="E8:74"/>
<span class="p1">(<span class="default"><a href="catalg_1.html#K11" title="CATALG_1:func.11">Upsilon</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K9" title="CATALG_1:func.9">homsym</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c15">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="catalg_1.html#E6:74"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="catalg_1.html#T30" target="_self" title="CATALG_1:th.30">Th30</a></i>;<br/>
<a NAME="E9:74"/><b>then </b><i><font color="Green" title="E42">A4</font></i>: 
( the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c2">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K11" title="CATALG_1:func.11">Upsilon</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c2">C2</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K11" title="CATALG_1:func.11">Upsilon</a> <font color="Maroon" title="c3">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K12" title="CATALG_1:func.12">Psi</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font> &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c2">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K11" title="CATALG_1:func.11">Upsilon</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c15">b</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="catalg_1.html#E2:74"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="catalg_1.html#E6:74"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="catalg_1.html#D15" target="_self" title="CATALG_1:def.15">Def15</a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<a NAME="E10:74"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font></span>)</span>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K16" title="CAT_1:func.16">hom</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>
 
<b>by </b><i><a class="txt" href="catalg_1.html#E7:74"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="cat_1.html#D25" title="CAT_1:def.25">CAT_1:def 25</a></i>;<br/>
<b>hence </b><a NAME="E11:74"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font></span>)</span>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c1">C1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K13" title="CATALG_1:func.13">MSAlg</a> <font color="Maroon" title="c2">C2</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C1</font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K11" title="CATALG_1:func.11">Upsilon</a> <font color="Maroon" title="c3">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K12" title="CATALG_1:func.12">Psi</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="catalg_1.html#E9:74"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/>


</div>
