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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>assume </b><a NAME="E1:27"/><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> )
 ;<br/>

<b>then consider </b><font color="Maroon">F0</font> being   <a href="ringcat1.html#NM1">RingMorphism</a><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E32</font></i>: 
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">F0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">F0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">F</font> =  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">F0</font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">F0</font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">F0</font> #) as   <a href="ringcat1.html#NM1">RingMorphism</a> <b>by </b><i/>;<br/>
<a NAME="E5:27"/>
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp;  <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> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">G0</font> being   <a href="ringcat1.html#NM1">RingMorphism</a><b> such that </b><br/><a NAME="E8:27"/><i><font color="Green">E34</font></i>: 
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">G0</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">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G3</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> =  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">G0</font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">G0</font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">G0</font> #) as   <a href="ringcat1.html#NM1">RingMorphism</a> <b>by </b><i/>;<br/>
<a NAME="E10:27"/>
(  <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> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G2</font>,<font color="Maroon">G3</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">g</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="E13:27"/><i><font color="Green">E35</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">g</font> #)
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">f</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="E15:27"/><i><font color="Green">E36</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">f</font> #)
 <b>by </b><i>, </i>;<br/>
<a NAME="E16:27"/>
 <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>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:27"/><b>then </b>
<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">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> #)
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E18:27"/><b>then </b>
(  <a href="ringcat1.html#K1" target="_self">dom</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">G1</font> &amp;  <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> )
 
;<br/>
<b>hence </b><a NAME="E19:27"/>
<font color="Maroon">G1</font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">G3</font>
 <b>by </b><i/>;<br/>


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