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<a NAME="E1:44_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:44_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">T</font> )</span><br/>

<b>assume </b><a NAME="E1:44_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">T</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:44_1_1_1"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="index_1.html#E3:44_1_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:44_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c6">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="ref" href="index_1.html#D9" target="_self" title="INDEX_1:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E6:44_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="index_1.html#E3:44_1_1_1"><i><font color="Green" title="E17">A2</font></i></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>then reconsider </b><font color="Maroon" title="c4">I1</font> = <font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">C</font>,the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">T</font> <b>by </b><i><a class="txt" href="index_1.html#E1:44_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> $1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> $1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> $1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>) <b>-&gt; </b>   <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 <font color="Maroon" title="c3">T</font> = <font color="Maroon" title="c4">I1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<div><i><font color="Green" title="E17">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>) is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">T</font> &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">T</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>) holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f</font>) &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font>) ) ) )</span><br/><b>thus </b><a NAME="E1:44_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>) is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="ref" href="index_1.html#D9" target="_self" title="INDEX_1:def.9">Def9</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" title="b1">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">T</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>) holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f</font>) &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font>) )</span><br/><b>let </b><font color="Maroon" title="c6">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>) implies (  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f</font>) &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font>) ) )</span><br/><b>assume </b><a NAME="E2:44_1_1_2"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f</font>) &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font>) )</span><br/><b>hence </b> <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>) <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T13" title="CAT_5:th.13">CAT_5:13</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>( <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f</font>)
<b>by </b><i><a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font>)</span><br/><b>thus </b> <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">g</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f</font>) <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a> 
<b>by </b><i><a class="txt" href="index_1.html#E2:44_1_1_2"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="cat_5.html#T13" title="CAT_5:th.13">CAT_5:13</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>( <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font>)
<b>by </b><i><a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</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>
<div><i><font color="Green" title="E18">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>( <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">a</font>)</span><br/><a NAME="E1:44_1_1_3"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>;<br/><b>hence </b><font color="Maroon">H<sub>1</sub></font>( <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">a</font>) = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">I1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">I1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="index_1.html#T6" target="_self" title="INDEX_1:th.6">Th6</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">a</font>)
<b>by </b><i><a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></i>
;<br/> <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>
<div><i><font color="Green" title="E19">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">T</font>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f1</font>) &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f2</font>) &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f1</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">g1</font></span><br/><b>let </b><font color="Maroon" title="c7">g1</font>, <font color="Maroon" title="c8">g2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f1</font>) &amp; <font color="Maroon" title="c8">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f2</font>) &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f1</font> implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">g1</font> )</span><br/><b>assume </b><a NAME="E1:44_1_1_4"/><i><font color="Green" title="E20">A7</font></i>: 
( <font color="Maroon" title="c7">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">f1</font>) &amp; <font color="Maroon" title="c8">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f2</font>) &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">g1</font></span><br/><a NAME="E2:44_1_1_4"/><i><font color="Green" title="E21">A8</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c5">f1</font> is    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c6">f2</font> is    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">f2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="index_1.html#T4" target="_self" title="INDEX_1:th.4">Th4</a></i>;<br/><a NAME="E3:44_1_1_4"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f1</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">f2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c5">f1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="index_1.html#E1:44_1_1_4"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="index_1.html#T6" target="_self" title="INDEX_1:th.6">Th6</a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_4"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">g1</font>
 <b>by </b><i><a class="txt" href="index_1.html#E1:44_1_1_4"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="index_1.html#E2:44_1_1_4"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</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="E7:44_1_1"/>
 ex <font color="Olive" title="b1">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T</font> st <br/>for <font color="Olive" title="b2">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">f</font>)
 <b>from </b><i><a class="ref" href="cat_5.html#S3" title="CAT_5:sch.3">CAT_5:sch 3</a>(<a class="txt" href="index_1.html#E4:44_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="index_1.html#E5:44_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="index_1.html#E6:44_1_1"><i><font color="Green" title="E19">A6</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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