<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ringcat1.html#NM1">RingMorphism</a><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E19</font></i>: 
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="ringcat1.html#D8" target="_self">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>4</sub></font> #) as   <a href="ringcat1.html#NM1">RingMorphism</a> <b>by </b><i><a class="txt" href="ringcat1.html#E13"><i><font color="Green">Lemma6</font></i></a></i>;<br/>
<a NAME="E5:27"/>
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E3:27"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="ringcat1.html#E1:27"><i><font color="Green">E18</font></i></a>, <a class="ref" href="ringcat1.html#D9" target="_self">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="ringcat1.html#NM1">RingMorphism</a><b> such that </b><br/><a NAME="E8:27"/><i><font color="Green">E20</font></i>: 
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:27"><i><font color="Green">E18</font></i></a>, <a class="ref" href="ringcat1.html#D8" target="_self">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>7</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>7</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>7</sub></font> #) as   <a href="ringcat1.html#NM1">RingMorphism</a> <b>by </b><i><a class="txt" href="ringcat1.html#E13"><i><font color="Green">Lemma6</font></i></a></i>;<br/>
<a NAME="E10:27"/>
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E8:27"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="ringcat1.html#E1:27"><i><font color="Green">E18</font></i></a>, <a class="ref" href="ringcat1.html#D9" target="_self">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E13:27"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> #)
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:27"><i><font color="Green">E18</font></i></a>, <a class="txt" href="ringcat1.html#E23"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E15:27"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font> #)
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:27"><i><font color="Green">E18</font></i></a>, <a class="txt" href="ringcat1.html#E23"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
 <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="ringcat1.html#E8:27"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
 <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="ringcat1.html#E3:27"><i><font color="Green">E19</font></i></a></i>
;<br/>
<a NAME="E17:27"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> #)
 
<b>by </b><i><a class="txt" href="ringcat1.html#E13:27"><i><font color="Green">E21</font></i></a>, <a class="txt" href="ringcat1.html#E15:27"><i><font color="Green">E22</font></i></a>, <a class="ref" href="ringcat1.html#D10" target="_self">Def10</a></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">c<sub>8</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E19:27"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="ringcat1.html#D8" target="_self">Def8</a></i>;<br/>


</div>
