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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><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="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green">E9</font></i>: 
 <a href="modcat_1.html#R1" target="_self">GO</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E10</font></i>: 
 <a href="modcat_1.html#R1" target="_self">GO</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:14"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:14"/><i><font color="Green">E12</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">b<sub>1</sub></font> #) &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E1:14"><i><font color="Green">E9</font></i></a>, <a class="ref" href="modcat_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:14"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E8:14"/><i><font color="Green">E14</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">b<sub>1</sub></font> #) &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E2:14"><i><font color="Green">E10</font></i></a>, <a class="ref" href="modcat_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:14"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>9</sub></font> #) &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E5:14"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being  <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:14"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>10</sub></font> #) &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E8:14"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E13:14"/>
(  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>9</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>10</sub></font> #) &amp; the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="modcat_1.html#E4:14"><i><font color="Green">E11</font></i></a>, <a class="txt" href="modcat_1.html#E7:14"><i><font color="Green">E13</font></i></a>, <a class="txt" href="modcat_1.html#E10:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="modcat_1.html#E12:14"><i><font color="Green">E16</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E14:14"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="modcat_1.html#E10:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="modcat_1.html#E12:14"><i><font color="Green">E16</font></i></a></i>;<br/>


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