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<b>consider </b><font color="Maroon">O</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">M</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">O</font>, <font color="Maroon">c</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">O</font>, <font color="Maroon">p</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">M</font>, <font color="Maroon">i</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">O</font>,<font color="Maroon">M</font>, <font color="Maroon">C</font> being    <a href="cat_1.html#L1" target="_self">CatStr</a> <b> such that </b><br/><a NAME="E2:27_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">o</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">d</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">m</font> <a href="cat_1.html#K7" target="_self">.--&gt;</a> <font color="Maroon">o</font> &amp; <font color="Maroon">c</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">m</font> <a href="cat_1.html#K7" target="_self">.--&gt;</a> <font color="Maroon">o</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>,<font color="Maroon">m</font> <a href="cat_1.html#K1" target="_self">.--&gt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">i</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">o</font> <a href="cat_1.html#K7" target="_self">.--&gt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1" target="_self">CatStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">o</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="cat_1.html#K7" target="_self">.--&gt;</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="cat_1.html#K7" target="_self">.--&gt;</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">m</font> <a href="cat_1.html#K1" target="_self">.--&gt;</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="cat_1.html#K7" target="_self">.--&gt;</a> <font color="Maroon">m</font></span>)</span> #) )
 ;<br/>
<a NAME="E3:27_1_1"/>
( ( 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" target="_self">Morphisms</a> of <font color="Maroon">C</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> iff the <a href="cat_1.html#U3" target="_self">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" target="_self">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" target="_self">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U3" target="_self">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" target="_self">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" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3" target="_self">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" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4" target="_self">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" target="_self">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U3" target="_self">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" target="_self">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" target="_self">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" target="_self">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" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</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" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">h</font>,<font color="Olive">g</font></span><a href="tarski.html#K4">]</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" target="_self">Objects</a> of <font color="Maroon">C</font> holds <br/> ( the <a href="cat_1.html#U3" target="_self">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" target="_self">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" target="_self">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" target="_self">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" target="_self">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U4" target="_self">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" target="_self">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#U6" target="_self">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="tarski.html#K4">]</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" target="_self">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U3" target="_self">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" target="_self">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">g</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6" target="_self">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="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) ) ) )
 
<b>by </b><i><a class="ref" href="cat_1.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>hence </b><a NAME="E4:27_1_1"/>
 <a href="cat_1.html#G1" target="_self">CatStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">o</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="cat_1.html#K7" target="_self">.--&gt;</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="cat_1.html#K7" target="_self">.--&gt;</a> <font color="Maroon">o</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font>,<font color="Maroon">m</font> <a href="cat_1.html#K1" target="_self">.--&gt;</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="cat_1.html#K7" target="_self">.--&gt;</a> <font color="Maroon">m</font></span>)</span> #) is  <a href="cat_1.html#V1" target="_self">strict</a> <a href="cat_1.html#NM4">Category</a>
 <b>by </b><i><a class="ref" href="cat_1.html#T4" target="_self">Th4</a>, </i>;<br/>


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