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<b>let </b><font color="Maroon">C</font> be    <a href="cat_1.html#L1">CatStr</a> , <font color="Maroon">D</font> be    <a href="cat_1.html#L1">CatStr</a> ;<br/>

<b>assume </b><a NAME="E1:3"/><i><font color="Green">E17</font></i>: 
 <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> #) <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> #)
 ;<br/>

<b>assume </b><a NAME="E2:3"/>
( ( for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> iff the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> ) ) &amp; ( for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> ) ) &amp; ( for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">h</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">h</font>,<font color="Olive">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> ) &amp; ( for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font> holds <br/> ( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; ( for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/>the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> ) &amp; ( for <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/>the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a><br/>

<b>hence </b><a NAME="E3:3"/>
( ( for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> iff the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> ) ) &amp; ( for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font>  st the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> ) ) &amp; ( for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font>  st the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">h</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">h</font>,<font color="Olive">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> ) &amp; ( for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font> holds <br/> ( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; ( for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font>  st the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/>the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> ) &amp; ( for <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font>  st the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/>the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) ) ) )
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a><br/>


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