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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="ringcat1.html#V2" target="_self">strict</a> <a href="ringcat1.html#NM1">RingMorphism</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:26_1_2"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="funcsdom.html#NM1">Ring</a><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font><br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> st  <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>1</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> #) &amp;  <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>2</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font> #) holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>5</sub></font></span>)</span> #)
 <b>and </b><br/><a NAME="E2:26_1_2"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="funcsdom.html#NM1">Ring</a><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font><br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> st  <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>1</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> #) &amp;  <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>2</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font> #) holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>5</sub></font></span>)</span> #)
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="funcsdom.html#NM1">Ring</a><b> such that </b><br/><a NAME="E4:26_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:26_1_2"/><i><font color="Green">E20</font></i>: 
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E6:26_1_2"/><i><font color="Green">E21</font></i>: 
 <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>1</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>1</sub></font> #) is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</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>1</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>1</sub></font> #) as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="ringcat1.html#E6:26_1_2"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="funcsdom.html#NM1">Ring</a><b> such that </b><br/><a NAME="E9:26_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="ringcat1.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E10:26_1_2"/><i><font color="Green">E22</font></i>: 
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="ringcat1.html#K2" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E11:26_1_2"/><i><font color="Green">E23</font></i>: 
 <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>2</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>2</sub></font> #) is    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</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>2</sub></font>,the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>2</sub></font> #) as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="ringcat1.html#E11:26_1_2"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="ringcat1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="ringcat1.html#E1:26"><i><font color="Green">E16</font></i></a>, <a class="txt" href="ringcat1.html#E5:26_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="ringcat1.html#E10:26_1_2"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E15:26_1_2"/><i><font color="Green">E24</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>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>12</sub></font> #)
 <b>by </b><i><a class="txt" href="ringcat1.html#E4:26_1_2"><i><font color="Green">E19</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>13</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E17:26_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font> #)
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:26"><i><font color="Green">E16</font></i></a>, <a class="txt" href="ringcat1.html#E5:26_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="ringcat1.html#E10:26_1_2"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> = 
 <a href="ringcat1.html#G1" target="_self">RingMorphismStr</a>(# <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> #)
<b>by </b><i><a class="txt" href="ringcat1.html#E1:26_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="ringcat1.html#E15:26_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="ringcat1.html#E17:26_1_2"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="ringcat1.html#E2:26_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="ringcat1.html#E15:26_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="ringcat1.html#E17:26_1_2"><i><font color="Green">E25</font></i></a></i>
;<br/>


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