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<b>let </b><font color="Maroon">G1</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">G3</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">G4</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G2</font>,<font color="Maroon">G3</font>;<br/><b>let </b><font color="Maroon">h</font> be    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G3</font>,<font color="Maroon">G4</font>;<br/>













<b>assume </b><a NAME="E1:32"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">G1</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">G2</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">G3</font> &amp; <font color="Maroon">G3</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">G4</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">f0</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font><b> such that </b><br/><a NAME="E3:32"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">G1</font>,<font color="Maroon">G2</font>,<font color="Maroon">f0</font> #)
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g0</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G2</font>,<font color="Maroon">G3</font><b> such that </b><br/><a NAME="E5:32"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">G2</font>,<font color="Maroon">G3</font>,<font color="Maroon">g0</font> #)
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">h0</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G3</font>,<font color="Maroon">G4</font><b> such that </b><br/><a NAME="E7:32"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">G3</font>,<font color="Maroon">G4</font>,<font color="Maroon">h0</font> #)
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:32"/><i><font color="Green">E36</font></i>: 
(  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> &amp;  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G3</font> &amp;  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G3</font> )
 
<b>by </b><i><a class="ref" href="ringcat1.html#D2" target="_self">Def2</a>, <a class="ref" href="ringcat1.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E9:32"/><b>then </b><i><font color="Green">E37</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>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">G1</font>,<font color="Maroon">G3</font>,<span class="p1">(<span class="default"><font color="Maroon">g0</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f0</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="ringcat1.html#D2" target="_self">Def2</a>, <a class="ref" href="ringcat1.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E10:32"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">h</font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">G2</font>,<font color="Maroon">G4</font>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g0</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="ringcat1.html#T3" target="_self">Th3</a>, , , </i>;<br/>
<a NAME="E11:32"/><i><font color="Green">E65</font></i>: 
(  <a href="ringcat1.html#K2" target="_self">cod</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> <font color="Maroon">G3</font> &amp;  <a href="ringcat1.html#K1" target="_self">dom</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="hidden.html#R1">=</a> <font color="Maroon">G2</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then </b><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="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">G1</font>,<font color="Maroon">G4</font>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">g0</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f0</font></span>)</span></span>)</span> #)
<b>by </b><i>, , , </i>
<br/>.= 
 <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">G1</font>,<font color="Maroon">G4</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g0</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f0</font></span>)</span> #)
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<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#D2" target="_self">Def2</a>, , , , </i>
;<br/>
<b>hence </b><a NAME="E13:32"/>
<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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