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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="cat_4.html#K23" title="CAT_4:func.23">c1Cat*</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font>) is  <a href="cat_1.html#V2" title="CAT_1:attr.2">Category-like</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">C</font> =  <a href="cat_4.html#K23" title="CAT_4:func.23">c1Cat*</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font>);<br/>
<span class="kw">set </span><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#K23" title="CAT_4:func.23">c1Cat*</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">CD</font> =  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">c1Cat*</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">CC</font> =  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">c1Cat*</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>;<br/>
<span class="kw">set </span><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#K23" title="CAT_4:func.23">c1Cat*</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:68"/>
 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#K23" title="CAT_4:func.23">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="relset_1.html#K1" title="RELSET_1:func.1">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#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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> )
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D5" title="CAT_1:def.5">CAT_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="68_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K23" title="CAT_4:func.23">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="relset_1.html#K1" title="RELSET_1:func.1">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#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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="E1:68_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K23" title="CAT_4:func.23">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E2:68_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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:68_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="relset_1.html#K1" title="RELSET_1:func.1">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#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E2:68"/>
 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#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="68_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K23" title="CAT_4:func.23">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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="E1:68_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#K23" title="CAT_4:func.23">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T77" title="FUNCOP_1:th.77">FUNCOP_1:77</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">c1Cat*</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:68_2"/>
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T50" title="FUNCT_2:th.50">FUNCT_2:50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:68_2"/>
(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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 class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T50" title="FUNCT_2:th.50">FUNCT_2:50</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E3:68"/>
 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#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="68_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K23" title="CAT_4:func.23">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="E1:68_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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T77" title="FUNCOP_1:th.77">FUNCOP_1:77</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">c1Cat*</a> (<font color="Maroon" title="c1">o</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:68_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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T77" title="FUNCOP_1:th.77">FUNCOP_1:77</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:68_3"/>
(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><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#K23" title="CAT_4:func.23">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="E4:68"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:68"/>
(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T50" title="FUNCT_2:th.50">FUNCT_2:50</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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/>

<span class="kw">thus </span><a NAME="E6:68"/>
 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#K23" title="CAT_4:func.23">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#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="68_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K23" title="CAT_4:func.23">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="E1:68_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:68_4"/>
(  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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 class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T77" title="FUNCOP_1:th.77">FUNCOP_1:77</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><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#K23" title="CAT_4:func.23">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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/>

<span class="kw">assume </span><a NAME="E7:68"/>
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" title="CAT_4:func.23">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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="E8:68"/>
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:68"/>
 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#K23" title="CAT_4:func.23">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#K23" title="CAT_4:func.23">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T77" title="FUNCOP_1:th.77">FUNCOP_1:77</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
