<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green" title="E9">A1</font></i>: 
 <a href="modcat_1.html#R1" title="MODCAT_1:pred.1">GO</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="modcat_1.html#R1" title="MODCAT_1:pred.1">GO</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y2</font>,<font color="Maroon" title="c1">R</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c6">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:14"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E5:14"/><i><font color="Green" title="E12">A4</font></i>: 
 ex <font color="Olive" title="b1">G</font> being <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a> <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Olive" title="b1">G</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Olive" title="b1">G</font> #) &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Olive" title="b1">G</font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="modcat_1.html#D5" target="_self" title="MODCAT_1:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">b1</font>, <font color="Maroon" title="c8">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:14"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E8:14"/><i><font color="Green" title="E14">A6</font></i>: 
 ex <font color="Olive" title="b1">G</font> being <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a> <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Olive" title="b1">G</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Olive" title="b1">G</font> #) &amp; <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Olive" title="b1">G</font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="modcat_1.html#D5" target="_self" title="MODCAT_1:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">G1</font> being  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a> <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E10:14"/><i><font color="Green" title="E15">A7</font></i>: 
( <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G1</font> &amp; <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">G1</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c9">G1</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c9">G1</font> #) &amp; <font color="Maroon" title="c6">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c9">G1</font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E5:14"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">G2</font> being  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a> <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E12:14"/><i><font color="Green" title="E16">A8</font></i>: 
( <font color="Maroon" title="c4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">G2</font> &amp; <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">G2</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c10">G2</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c10">G2</font> #) &amp; <font color="Maroon" title="c8">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c10">G2</font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E8:14"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/>
<a NAME="E13:14"/>
(  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">G1</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c9">G1</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c9">G1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">G2</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c10">G2</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c10">G2</font> #) &amp; the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c9">G1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c10">G2</font> )
 
<b>by </b><i><a class="txt" href="modcat_1.html#E4:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="modcat_1.html#E7:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="modcat_1.html#E10:14"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="modcat_1.html#E12:14"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E14:14"/>
<font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y2</font>
 <b>by </b><i><a class="txt" href="modcat_1.html#E10:14"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="modcat_1.html#E12:14"><i><font color="Green" title="E16">A8</font></i></a></i>;<br/>


</div>
