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

<b>let </b><font color="Maroon" title="c21">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c22">P</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c21">B</font>;<br/><b>let </b><font color="Maroon" title="c23">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c21">B</font>;<br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c21">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="realset1.html#K5" title="REALSET1:func.5">{</a><span class="default"><font color="Maroon" title="c5">nd</font></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span> &amp; <font color="Maroon" title="c23">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">nm</font> &amp; <font color="Maroon" title="c22">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">om1</font> <a href="realset1.html#K4" title="REALSET1:func.4">!</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">nd</font> )
 ;<br/>

<a NAME="E2:26"/><i><font color="Green" title="E49">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">B</font>,<font color="Maroon" title="c21">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">B</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E3:26"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">B</font>,<font color="Maroon" title="c21">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E4:26"/><b>then </b>
<font color="Maroon" title="c22">P</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">P</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E2:26"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:26"/><b>then </b><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c22">P</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="realset2.html#E2:26"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green" title="E40">Lm35</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:26"/><i><font color="Green" title="E51">A4</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c21">B</font> holds  <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span>,<font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">a</font>, <font color="Maroon" title="c25">b</font>, <font color="Maroon" title="c26">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c21">B</font>;<br/>

<a NAME="E1:26_1"/>
( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c25">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c26">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )
 
<b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green" title="E40">Lm35</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:26_1"/>
<font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span>)</span>,<font color="Maroon" title="c26">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c24">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c25">b</font>,<font color="Maroon" title="c26">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="realset2.html#E5:26"><i><font color="Green" title="E50">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:26"/><i><font color="Green" title="E52">A5</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c21">B</font> holds <br/> ( <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c23">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">e</font>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c21">B</font>;<br/>

<a NAME="E1:26_2"/>
<font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green" title="E40">Lm35</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:26_2"/>
( <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c23">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">a</font> &amp; <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">e</font>,<font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">a</font> )
 <b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green" title="E40">Lm35</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:26"/><i><font color="Green" title="E53">A6</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c21">B</font>  ex <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c21">B</font> st <br/>( <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">e</font> &amp; <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">e</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c21">B</font>;<br/>

<b>take </b>
<font color="Maroon" title="c23">e</font>
;<br/>

<b>thus </b><a NAME="E1:26_3"/>
( <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c23">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">e</font> &amp; <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">e</font>,<font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">e</font> )
 <b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green" title="E40">Lm35</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:26"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c21">B</font> holds  <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">a</font>, <font color="Maroon" title="c25">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c21">B</font>;<br/>

<a NAME="E1:26_4"/>
( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c25">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )
 
<b>by </b><i><a class="txt" href="realset2.html#E1:26"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="realset2.html#E57"><i><font color="Green" title="E40">Lm35</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:26_4"/>
<font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c25">b</font>,<font color="Maroon" title="c24">a</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:26"/>
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c21">B</font>,<font color="Maroon" title="c22">P</font>,<font color="Maroon" title="c23">e</font> #) is   <a href="realset2.html#NM1" title="REALSET2:NM.1">Group</a>
 <b>by </b><i><a class="txt" href="realset2.html#E6:26"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="realset2.html#E7:26"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="realset2.html#E8:26"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="realset2.html#D1" target="_self" title="REALSET2:def.1">Def1</a>, <a class="ref" href="realset2.html#D2" target="_self" title="REALSET2:def.2">Def2</a>, <a class="ref" href="realset2.html#D3" target="_self" title="REALSET2:def.3">Def3</a>, <a class="ref" href="realset2.html#D4" target="_self" title="REALSET2:def.4">Def4</a></i>;<br/>


</div>
