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<b>let </b><font color="Maroon">f</font> be  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>, <font color="Maroon">g</font> be  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>, <font color="Maroon">h</font> be  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>;<br/>

<b>assume </b><a NAME="E1:33"/><i><font color="Green">E31</font></i>: 
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font> &amp;  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font> )
 ;<br/>

<b>set </b><font color="Maroon">G1</font> =  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">G2</font> =  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">G3</font> =  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font>;<br/>
<b>set </b><font color="Maroon">G4</font> =  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">h</font>;<br/>
<a NAME="E2:33"/><i><font color="Green">E32</font></i>: 
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">f</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a>  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a>  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a>  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">h</font> )
 
<b>by </b><i><a class="ref" href="ringcat1.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">f</font>, <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font>, <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font> <b>by </b><i><a class="ref" href="ringcat1.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h</font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font>, <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">h</font> <b>by </b><i><a class="ref" href="ringcat1.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E6:33"/>
<font color="Maroon">h'</font> <a href="ringcat1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="ringcat1.html#K6" target="_self">*</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="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">g'</font></span>)</span> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">f'</font>
 
<b>by </b><i><a class="ref" href="ringcat1.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>hence </b><a NAME="E7:33"/>
<font color="Maroon">h</font> <a href="ringcat1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="ringcat1.html#K6" target="_self">*</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="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">f</font>
 ;<br/>


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