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<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>, <font color="Maroon">c<sub>4</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <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>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <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>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>













<b>assume </b><a NAME="E1:32"/><i><font color="Green">E23</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> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</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="E3:32"/><i><font color="Green">E24</font></i>: 
<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>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> #)
 <b>by </b><i><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>9</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="E5:32"/><i><font color="Green">E25</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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> #)
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:32"><i><font color="Green">E23</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>10</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E7:32"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> #)
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:32"><i><font color="Green">E23</font></i></a>, <a class="txt" href="ringcat1.html#E23"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<a NAME="E8:32"/><i><font color="Green">E27</font></i>: 
(  <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> &amp;  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>6</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>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <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>3</sub></font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E3:32"><i><font color="Green">E24</font></i></a>, <a class="txt" href="ringcat1.html#E5:32"><i><font color="Green">E25</font></i></a>, <a class="txt" href="ringcat1.html#E7:32"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E9:32"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>6</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>9</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> #)
 
<b>by </b><i><a class="txt" href="ringcat1.html#E3:32"><i><font color="Green">E24</font></i></a>, <a class="txt" href="ringcat1.html#E5:32"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ringcat1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E10:32"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <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>2</sub></font>,<font color="Maroon">c<sub>4</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>9</sub></font></span>)</span> #)
 
<b>by </b><i><a class="txt" href="ringcat1.html#E5:32"><i><font color="Green">E25</font></i></a>, <a class="txt" href="ringcat1.html#E7:32"><i><font color="Green">E26</font></i></a>, <a class="txt" href="ringcat1.html#E8:32"><i><font color="Green">E27</font></i></a>, <a class="ref" href="ringcat1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E11:32"/><i><font color="Green">E30</font></i>: 
(  <a href="ringcat1.html#K2" target="_self">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</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> &amp;  <a href="ringcat1.html#K1" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="ringcat1.html#E8:32"><i><font color="Green">E27</font></i></a>, <a class="ref" href="ringcat1.html#T11" target="_self">Th11</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>7</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
 <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> #)
<b>by </b><i><a class="txt" href="ringcat1.html#E7:32"><i><font color="Green">E26</font></i></a>, <a class="txt" href="ringcat1.html#E8:32"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ringcat1.html#E9:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ringcat1.html#D10" target="_self">Def10</a></i>
<br/>.= 
 <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></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">c<sub>7</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="ringcat1.html#E3:32"><i><font color="Green">E24</font></i></a>, <a class="txt" href="ringcat1.html#E8:32"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ringcat1.html#E10:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ringcat1.html#E11:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="ringcat1.html#D10" target="_self">Def10</a></i>
;<br/>
<b>hence </b><a NAME="E13:32"/>
<font color="Maroon">c<sub>7</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="ringcat1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


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