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<b>let </b><font color="Maroon" title="c1">o</font>, <font color="Maroon" title="c2">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font> is <a href="cat_1.html#V3" title="CAT_1:attr.3">Category-like</a></span><br/>



<b>set </b><font color="Maroon" title="c3">C</font> =  <a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font>;<br/>
<b>set </b><font color="Maroon" title="c4">CP</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">CD</font> = the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>;<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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">CI</font> = the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<b>thus </b><a NAME="E1:10"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> iff the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> )
  <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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> ) ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>3</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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> or  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> holds <br/> ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> iff the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> )</span><br/>

<a NAME="E1:10_1"/>
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c2">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:10_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> iff the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:10"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  st the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>3</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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> or  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> holds <br/> ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> implies ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> ) )</span><br/>

<a NAME="E1:10_2"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T9" title="CAT_1:th.9">CAT_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">gf</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:10_2"/>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">o</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">o</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">gf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">o</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">gf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">o</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T65" title="FUNCT_2:th.65">FUNCT_2:65</a></i>;<br/>
<b>hence </b><a NAME="E4:10_2"/>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> implies ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:10"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  st the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b1">f</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> holds <br/> ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c10">h</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> implies the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">h</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c9">g</font></span>)</span>,<font color="Maroon" title="c8">f</font> )</span><br/>

<a NAME="E1:10_3"/>
( the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T9" title="CAT_1:th.9">CAT_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">gf</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c12">hg</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c9">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:10_3"/>
( the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c11">gf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">hg</font>,<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T9" title="CAT_1:th.9">CAT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E4:10_3"/>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> implies the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">h</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c9">g</font></span>)</span>,<font color="Maroon" title="c8">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c8">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<a NAME="E4:10"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">o</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:10"/>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T65" title="FUNCT_2:th.65">FUNCT_2:65</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<b>thus </b><a NAME="E6:10"/>
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  st the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/>the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>  holds <br/>(  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> implies the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> )</span><br/>

<a NAME="E1:10_4"/>
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:10_4"/>
( the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> implies the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <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 <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T9" title="CAT_1:th.9">CAT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> )</span><br/>

<b>assume </b><a NAME="E7:10"/>
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font></span><br/>

<a NAME="E8:10"/>
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:10"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K5" title="CAT_4:func.5">c1Cat</a> <font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T9" title="CAT_1:th.9">CAT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
