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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</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">E33</font></i>: 
 <a href="modcat_1.html#R1" target="_self">GO</a> <font color="Maroon">x</font>,<font color="Maroon">y1</font>,<font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E37</font></i>: 
 <a href="modcat_1.html#R1" target="_self">GO</a> <font color="Maroon">x</font>,<font color="Maroon">y2</font>,<font color="Maroon">R</font>
 ;<br/>

<b>consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:14"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:14"/><i><font color="Green">E39</font></i>: 
 ex <font color="Olive">G</font> being <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Maroon">a1</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">G</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">G</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">G</font> #) &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Olive">G</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">b1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:14"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E8:14"/><i><font color="Green">E44</font></i>: 
 ex <font color="Olive">G</font> being <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> &amp; <font color="Maroon">b1</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">G</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Olive">G</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Olive">G</font> #) &amp; <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Olive">G</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">G1</font> being  <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E10:14"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">a1</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">G1</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G1</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">G1</font> #) &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">G1</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G2</font> being  <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E12:14"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">b1</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">G2</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G2</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">G2</font> #) &amp; <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">G2</font> )
 <b>by </b><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">G1</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G1</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">G1</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">G2</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">G2</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">G2</font> #) &amp; the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">G1</font> <a href="funct_2.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">G2</font> )
 
<b>by </b><i>, , , , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E14:14"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i>, </i>;<br/>


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