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( ( for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">UN</font></span>)</span> 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="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">UN</font></span>)</span> iff  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> ) ) &amp; ( for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">UN</font></span>)</span>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> holds <br/>(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Olive">f</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</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="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">UN</font></span>)</span>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">g</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> holds <br/><font color="Olive">h</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="cat_1.html#K4">*</a> <font color="Olive">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font> ) &amp; ( for <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">UN</font></span>)</span> holds <br/> (  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</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="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">UN</font></span>)</span>  st  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Olive">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> ) &amp; ( for <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">UN</font></span>)</span>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/><font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) ) ) )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E16"><i><font color="Green">Lemma49</font></i></a>, , <a class="txt" href="ringcat1.html#E17"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="ringcat1.html#E13"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:77_1_1"/>
 <a href="ringcat1.html#K18" target="_self">RingCat</a> <font color="Maroon">UN</font> is <a href="cat_1.html#V2">Category-like</a>
 <b>by </b><i><a class="ref" href="cat_1.html#T29">CAT_1:29</a></i>;<br/>


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