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<b>let </b><font color="Maroon" title="c21">F</font> be   <a href="realset2.html#NM2" title="REALSET2:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c22">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="c23">P</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c22">B</font>;<br/><b>let </b><font color="Maroon" title="c24">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c22">B</font>;<br/>







<a NAME="E1:37"/>
 ex <font color="Olive" title="b1">A</font> being non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">od</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">A</font> ex <font color="Olive" title="b3">nd</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">A</font> ex <font color="Olive" title="b4">om</font> being   <a href="realset1.html#M3" title="REALSET1:mode.3">DnT</a> of <font color="Olive" title="b3">nd</font>,<font color="Olive" title="b1">A</font> ex <font color="Olive" title="b5">nm</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">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="Olive" title="b3">nd</font></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span> st <br/>( <font color="Maroon" title="c21">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K1" title="REALSET2:func.1">field</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">od</font>,<font color="Olive" title="b4">om</font>,<font color="Olive" title="b3">nd</font>,<font color="Olive" title="b5">nm</font> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">od</font>,<font color="Olive" title="b3">nd</font> #) is   <a href="realset2.html#NM1" title="REALSET2:NM.1">Group</a> &amp; ( for <font color="Olive" title="b6">B</font> being 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/> for <font color="Olive" title="b7">P</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b6">B</font><br/> for <font color="Olive" title="b8">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b6">B</font>  st <font color="Olive" title="b6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b3">nd</font></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span> &amp; <font color="Olive" title="b8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">nm</font> &amp; <font color="Olive" title="b7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">om</font> <a href="realset1.html#K4" title="REALSET1:func.4">!</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">nd</font> holds <br/> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Olive" title="b6">B</font>,<font color="Olive" title="b7">P</font>,<font color="Olive" title="b8">e</font> #) is   <a href="realset2.html#NM1" title="REALSET2:NM.1">Group</a> ) &amp; ( for <font color="Olive" title="b6">x</font>, <font color="Olive" title="b7">y</font>, <font color="Olive" title="b8">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">A</font> holds <br/> ( <font color="Olive" title="b4">om</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">od</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b7">y</font>,<font color="Olive" title="b8">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">od</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">om</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">x</font>,<font color="Olive" title="b7">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">om</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">x</font>,<font color="Olive" title="b8">z</font></span>)</span> &amp; <font color="Olive" title="b4">om</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">od</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">x</font>,<font color="Olive" title="b7">y</font></span>)</span>,<font color="Olive" title="b8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">od</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">om</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b6">x</font>,<font color="Olive" title="b8">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">om</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b7">y</font>,<font color="Olive" title="b8">z</font></span>)</span> ) ) )
 
<b>by </b><i><a class="ref" href="realset2.html#D8" target="_self" title="REALSET2:def.8">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c25">A</font> being  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c26">od</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c25">A</font>, <font color="Maroon" title="c27">nd</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c25">A</font>, <font color="Maroon" title="c28">om</font> being    <a href="realset1.html#M3" title="REALSET1:mode.3">DnT</a> of <font color="Maroon" title="c27">nd</font>,<font color="Maroon" title="c25">A</font>, <font color="Maroon" title="c29">nm</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c25">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="c27">nd</font></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span><b> such that </b><br/><a NAME="E3:37"/><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c21">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K1" title="REALSET2:func.1">field</a> <font color="Maroon" title="c25">A</font>,<font color="Maroon" title="c26">od</font>,<font color="Maroon" title="c28">om</font>,<font color="Maroon" title="c27">nd</font>,<font color="Maroon" title="c29">nm</font>
 <b>and </b><br/><a NAME="E4:37"/><i><font color="Green" title="E58">A2</font></i>: 
for <font color="Olive" title="b1">B</font> being 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/> for <font color="Olive" title="b2">P</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">B</font><br/> for <font color="Olive" title="b3">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">B</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">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="c27">nd</font></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">nm</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">om</font> <a href="realset1.html#K4" title="REALSET1:func.4">!</a> <font color="Maroon" title="c25">A</font>,<font color="Maroon" title="c27">nd</font> holds <br/> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">e</font> #) is   <a href="realset2.html#NM1" title="REALSET2:NM.1">Group</a>
 ;<br/>
<a NAME="E5:37"/><i><font color="Green" title="E59">A3</font></i>: 
<font color="Maroon" title="c25">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="realset2.html#D9" target="_self" title="REALSET2:def.9">Def9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">od</font> = <font color="Maroon" title="c26">od</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c31">nd</font> = <font color="Maroon" title="c27">nd</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font> <b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="realset2.html#D9" target="_self" title="REALSET2:def.9">Def9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">om</font> = <font color="Maroon" title="c28">om</font> as    <a href="realset1.html#M3" title="REALSET1:mode.3">DnT</a> of <font color="Maroon" title="c31">nd</font>, <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font> <b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="realset2.html#D9" target="_self" title="REALSET2:def.9">Def9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">nm</font> = <font color="Maroon" title="c29">nm</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font></span>)</span> <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="c31">nd</font></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span> <b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="realset2.html#D9" target="_self" title="REALSET2:def.9">Def9</a></i>;<br/>
<a NAME="E10:37"/><i><font color="Green" title="E60">A4</font></i>: 
<font color="Maroon" title="c30">od</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="realset2.html#D10" target="_self" title="REALSET2:def.10">Def10</a></i>;<br/>
<a NAME="E11:37"/><i><font color="Green" title="E61">A5</font></i>: 
<font color="Maroon" title="c21">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K1" title="REALSET2:func.1">field</a> <span class="p1">(<span class="default"><a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="realset2.html#K3" title="REALSET2:func.3">odf</a> <font color="Maroon" title="c21">F</font></span>)</span>,<font color="Maroon" title="c32">om</font>,<font color="Maroon" title="c31">nd</font>,<font color="Maroon" title="c33">nm</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="realset2.html#D10" target="_self" title="REALSET2:def.10">Def10</a></i>;<br/>
<a NAME="E12:37"/><i><font color="Green" title="E62">A6</font></i>: 
<font color="Maroon" title="c31">nd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="realset2.html#E10:37"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="realset2.html#D11" target="_self" title="REALSET2:def.11">Def11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">om</font> = <font color="Maroon" title="c32">om</font> as    <a href="realset1.html#M3" title="REALSET1:mode.3">DnT</a> of  <a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font>, <a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font> <b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="realset2.html#E10:37"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="realset2.html#D11" target="_self" title="REALSET2:def.11">Def11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c35">nm</font> = <font color="Maroon" title="c33">nm</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font></span>)</span></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span> <b>by </b><i><a class="txt" href="realset2.html#E3:37"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="realset2.html#E10:37"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="realset2.html#D11" target="_self" title="REALSET2:def.11">Def11</a></i>;<br/>
<a NAME="E15:37"/><i><font color="Green" title="E63">A7</font></i>: 
<font color="Maroon" title="c34">om</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E11:37"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="realset2.html#E12:37"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="realset2.html#D12" target="_self" title="REALSET2:def.12">Def12</a></i>;<br/>
<a NAME="E16:37"/><b>then </b><i><font color="Green" title="E64">A8</font></i>: 
<font color="Maroon" title="c35">nm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K6" title="REALSET2:func.6">nmf</a> <font color="Maroon" title="c21">F</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E11:37"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="realset2.html#E12:37"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="realset2.html#D13" target="_self" title="REALSET2:def.13">Def13</a></i>;<br/>
<b>assume </b><a NAME="E17:37"/>
( <font color="Maroon" title="c22">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font></span>)</span></span><a href="realset1.html#K5" title="REALSET1:func.5">}</a></span> &amp; <font color="Maroon" title="c24">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="realset2.html#K6" title="REALSET2:func.6">nmf</a> <font color="Maroon" title="c21">F</font> &amp; <font color="Maroon" title="c23">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">omf</a> <font color="Maroon" title="c21">F</font></span>)</span> <a href="realset1.html#K4" title="REALSET1:func.4">!</a> <span class="p1">(<span class="default"><a href="realset2.html#K2" title="REALSET2:func.2">suppf</a> <font color="Maroon" title="c21">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">ndf</a> <font color="Maroon" title="c21">F</font></span>)</span> )
 ;<br/>

<b>hence </b><a NAME="E18:37"/>
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c22">B</font>,<font color="Maroon" title="c23">P</font>,<font color="Maroon" title="c24">e</font> #) is   <a href="realset2.html#NM1" title="REALSET2:NM.1">Group</a>
 <b>by </b><i><a class="txt" href="realset2.html#E4:37"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="realset2.html#E5:37"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="realset2.html#E12:37"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="realset2.html#E15:37"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="realset2.html#E16:37"><i><font color="Green" title="E64">A8</font></i></a></i>;<br/>


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