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

<b>set </b><font color="Maroon">A</font> =  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Olive">P</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Olive">Q</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Olive">Q</font></span>)</span> );<br/>
<a NAME="E1:54_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>;<br/>

<a NAME="E1:54_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon">P</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:54_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E5:54_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon">Q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:54_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E8:54_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font>, <font color="Maroon">Q</font> = <font color="Maroon">Q</font> as   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E10:54_1_1_1"/>
<font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a>  <a href="topalg_1.html#K1" target="_self">Loops</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span> <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E12:54_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span><b> such that </b><br/><a NAME="E3:54_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<b>take </b><font color="Maroon">G</font> =  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <span class="p2">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #);<br/>

<b>thus </b><a NAME="E4:54_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  ex <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">a</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Olive">P</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<font color="Olive">Q</font> &amp; the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2" target="_self">EqRel</a> <font color="Maroon">X</font>,<font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Olive">Q</font></span>)</span> ) ) )
 <b>by </b><i/>;<br/>


</div>
