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

<b>let </b><font color="Maroon" title="c17">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="c18">P</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c17">B</font>;<br/><b>let </b><font color="Maroon" title="c19">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c17">B</font>;<br/>





<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c17">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="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">nd</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">nm</font> &amp; <font color="Maroon" title="c18">P</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c16">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="E2:15"/><i><font color="Green" title="E27">A2</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="c17">B</font> holds  <font color="Maroon" title="c18">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">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="c18">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="c18">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="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">a</font>, <font color="Maroon" title="c21">b</font>, <font color="Maroon" title="c22">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c17">B</font>;<br/>

<a NAME="E1:15_1"/>
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,2 &amp; <font color="Maroon" title="c21">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,2 &amp; <font color="Maroon" title="c22">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,2 )
 
<b>by </b><i><a class="txt" href="realset2.html#E1:15"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="realset2.html#E28"><i><font color="Green" title="E17">Lm17</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="E2:15_1"/>
<font color="Maroon" title="c18">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a</font>,<font color="Maroon" title="c21">b</font></span>)</span>,<font color="Maroon" title="c22">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">P</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c21">b</font>,<font color="Maroon" title="c22">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="realset2.html#E1:15"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="realset2.html#E28"><i><font color="Green" title="E17">Lm17</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="E3:15"/><i><font color="Green" title="E28">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #) holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #)</span>)</span> <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="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #);<br/>

<a NAME="E1:15_2"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,2
 
<b>by </b><i><a class="txt" href="realset2.html#E1:15"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="realset2.html#E28"><i><font color="Green" title="E17">Lm17</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="E2:15_2"/>
<font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font>
 <b>by </b><i><a class="txt" href="realset2.html#E1:15"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="realset2.html#E28"><i><font color="Green" title="E17">Lm17</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="E4:15"/><i><font color="Green" title="E29">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #)  ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #) st <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #);<br/>

<b>take </b>
<font color="Maroon" title="c20">a</font>
;<br/>

<b>thus </b><a NAME="E1:15_3"/>
<font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #)
 <b>by </b><i><a class="txt" href="realset2.html#E1:15"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="realset2.html#E28"><i><font color="Green" title="E17">Lm17</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="E5:15"/><i><font color="Green" title="E30">A5</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b3">c</font></span>)</span>
 
<b>by </b><i><a class="txt" href="realset2.html#E2:15"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/>
<a NAME="E6:15"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #) holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">a</font>, <font color="Maroon" title="c21">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #);<br/>

<a NAME="E1:15_4"/>
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,2 &amp; <font color="Maroon" title="c21">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 1,2 )
 
<b>by </b><i><a class="txt" href="realset2.html#E1:15"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="realset2.html#E28"><i><font color="Green" title="E17">Lm17</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="E2:15_4"/>
<font color="Maroon" title="c20">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c21">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c20">a</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:15"/>
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <font color="Maroon" title="c17">B</font>,<font color="Maroon" title="c18">P</font>,<font color="Maroon" title="c19">e</font> #) is   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a>
 <b>by </b><i><a class="txt" href="realset2.html#E3:15"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="realset2.html#E4:15"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="realset2.html#E5:15"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a>, <a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></i>;<br/>


</div>
