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<div class="add">

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <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="Olive">b<sub>1</sub></font> #) where B is    <a href="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ringcat1.html#V1" target="_self">linear</a> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <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>consider </b><font color="Maroon">c<sub>5</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="E3:49_1_1"/>
<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="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> #)
 <b>and </b><br/><a NAME="E4:49_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="ringcat1.html#V1" target="_self">linear</a>
 <b>by </b><i><a class="txt" href="ringcat1.html#E1:49"><i><font color="Green">E30</font></i></a>, <a class="txt" href="ringcat1.html#E22"><i><font color="Green">Lemma14</font></i></a></i>;<br/>
<a NAME="E5:49_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</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="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E7:49_1_1"/>
 <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>6</sub></font> #) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <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="Olive">b<sub>1</sub></font> #) where B is    <a href="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ringcat1.html#V1" target="_self">linear</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ringcat1.html#E4:49_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <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="Olive">b<sub>1</sub></font> #) where B is    <a href="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="ringcat1.html#V1" target="_self">linear</a> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E9:49_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is    <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:49_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>8</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="Olive">b<sub>1</sub></font> #) &amp; <font color="Olive">b<sub>1</sub></font> is <a href="ringcat1.html#V1" target="_self">linear</a> )
 
;<br/>
<b>hence </b><a NAME="E3:49_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> is    <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#E17"><i><font color="Green">Lemma10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:49_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> holds <font color="Olive">b<sub>1</sub></font> is    <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/>
<a NAME="E11:49_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> is    <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> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> is    <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>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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E3:49_1_1_2"/>
(  <a href="ringcat1.html#K1" target="_self">dom</a> <font color="Maroon">c<sub>9</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>9</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#E1:49"><i><font color="Green">E30</font></i></a>, <a class="ref" href="ringcat1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E4:49_1_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
( the <a href="ringcat1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="ringcat1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<a NAME="E5:49_1_1_2"/><i><font color="Green">E36</font></i>: 
the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>9</sub></font> is <a href="ringcat1.html#V1" target="_self">linear</a>
 
<b>by </b><i><a class="txt" href="ringcat1.html#E15"><i><font color="Green">Lemma8</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = the <a href="ringcat1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>9</sub></font> as   <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>by </b><i><a class="txt" href="ringcat1.html#E4:49_1_1_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E7:49_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E9:49_1_1_2"/>
<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>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#E4:49_1_1_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:49_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="ringcat1.html#E5:49_1_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="ringcat1.html#M4" target="_self">RingMorphism_DOMAIN</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#E10:49_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ringcat1.html#T18" target="_self">Th18</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E13:49_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> iff <font color="Olive">b<sub>1</sub></font> is    <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#E9:49_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="ringcat1.html#E11:49_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div>
