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

<b>let </b><font color="Maroon" title="c17">F</font> be  non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="realset2.html#V1" title="REALSET2:attr.1">Field-like</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c18">a</font>, <font color="Maroon" title="c19">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>



<b>set </b><font color="Maroon" title="c20">B</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c21">P</font> = <span class="p1">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">omf</a> <font color="Maroon" title="c17">F</font></span>)</span> <a href="realset1.html#K4" title="REALSET1:func.4">!</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</font>),<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c22">e</font> =  <a href="realset2.html#K5" title="REALSET2:func.5">nmf</a> <font color="Maroon" title="c17">F</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">D</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</font>) <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">omf</a> <font color="Maroon" title="c17">F</font></span>)</span> <a href="realset1.html#K4" title="REALSET1:func.4">!</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</font>),<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="realset2.html#K5" title="REALSET2:func.5">nmf</a> <font color="Maroon" title="c17">F</font></span>)</span> #) as  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a> <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> <b>by </b><i><a class="ref" href="realset2.html#T2" target="_self" title="REALSET2:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">a</font> = <font color="Maroon" title="c18">a</font>, <font color="Maroon" title="c25">b</font> = <font color="Maroon" title="c19">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c23">D</font> ;<br/>
<a NAME="E3:34"/>
<span class="p1">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">omf</a> <font color="Maroon" title="c17">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</font>) <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</font>) <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</font>) <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="realset1.html#T11" title="REALSET1:th.11">REALSET1:11</a></i>;<br/>
<a NAME="E4:34"/><b>then </b><i><font color="Green" title="E32">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">omf</a> <font color="Maroon" title="c17">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</font>) <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</font>) <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</font>) <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:34"/><i><font color="Green" title="E33">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <span class="p1">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">omf</a> <font color="Maroon" title="c17">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c23">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">x</font>, <font color="Maroon" title="c27">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>

<a NAME="E1:34_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c27">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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"><span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</font>) <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">F</font>) <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">F</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<b>hence </b><a NAME="E2:34_1"/>
<span class="p1">(<span class="default"><a href="realset2.html#K4" title="REALSET2:func.4">omf</a> <font color="Maroon" title="c17">F</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c23">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c27">y</font>
 <b>by </b><i><a class="txt" href="realset2.html#E4:34"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c26">x</font> = <font color="Maroon" title="c24">a</font>, <font color="Maroon" title="c27">y</font> = <font color="Maroon" title="c25">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c17">F</font> ;<br/>
<font color="Maroon" title="c26">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c27">y</font> = 
<font color="Maroon" title="c24">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c25">b</font>
<b>by </b><i><a class="txt" href="realset2.html#E5:34"><i><font color="Green" title="E33">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c25">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c24">a</font>

<br/>.= 
<font color="Maroon" title="c27">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="realset2.html#E5:34"><i><font color="Green" title="E33">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E8:34"/>
<font color="Maroon" title="c18">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">a</font>
 ;<br/>


</div>
