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<b>let </b><font color="Maroon">UN</font> be   <a href="classes2.html#NM1">Universe</a>;<br/>

<b>set </b><font color="Maroon">X</font> =  <a href="ringcat1.html#K10" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">UN</font></span>)</span>;<br/>
<b>let </b><font color="Maroon">f</font> be   <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>, <font color="Maroon">g</font> be   <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>, <font color="Maroon">h</font> be   <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>;<br/>

<b>assume </b><a NAME="E1:75"/><i><font color="Green">E31</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font>, <font color="Maroon">g'</font> = <font color="Maroon">g</font>, <font color="Maroon">h'</font> = <font color="Maroon">h</font> as  <a href="ringcat1.html#V2" target="_self">strict</a>  <a href="ringcat1.html#M3" target="_self">Element</a> of  <a href="ringcat1.html#K10" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">UN</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E3:75"/><i><font color="Green">E32</font></i>: 
(  <a href="ringcat1.html#K11" target="_self">dom</a> <font color="Maroon">h'</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K12" target="_self">cod</a> <font color="Maroon">g'</font> &amp;  <a href="ringcat1.html#K11" target="_self">dom</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K12" target="_self">cod</a> <font color="Maroon">f'</font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E17"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="ringcat1.html#E15"><i><font color="Green">Lemma48</font></i></a></i>;<br/>
<a NAME="E4:75"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">g'</font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">h'</font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E17"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="ringcat1.html#E15"><i><font color="Green">Lemma48</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">gf</font> = <font color="Maroon">g'</font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">f'</font>, <font color="Maroon">hg</font> = <font color="Maroon">h'</font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">g'</font> as  <a href="ringcat1.html#V2" target="_self">strict</a>  <a href="ringcat1.html#M3" target="_self">Element</a> of  <a href="ringcat1.html#K10" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" target="_self">RingObjects</a> <font color="Maroon">UN</font></span>)</span> <b>by </b><i><a class="txt" href="ringcat1.html#E18"><i><font color="Green">Lemma52</font></i></a>, </i>;<br/>
<a NAME="E6:75"/><i><font color="Green">E35</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E17"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="ringcat1.html#E16"><i><font color="Green">Lemma49</font></i></a></i>;<br/>
<a NAME="E7:75"/><i><font color="Green">E36</font></i>: 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E17"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="ringcat1.html#E16"><i><font color="Green">Lemma49</font></i></a></i>;<br/>
<font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">h'</font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">gf</font>
<b>by </b><i><a class="ref" href="ringcat1.html#T5" target="_self">Th5</a>, , <a class="txt" href="ringcat1.html#E15"><i><font color="Green">Lemma48</font></i></a></i>
<br/>.= 
<font color="Maroon">hg</font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">f'</font>
<b>by </b><i><a class="txt" href="ringcat1.html#E18"><i><font color="Green">Lemma52</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="ringcat1.html#T5" target="_self">Th5</a>, , <a class="txt" href="ringcat1.html#E15"><i><font color="Green">Lemma48</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:75"/>
<font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>
 ;<br/>


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