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<span class="kw">set </span><font color="Maroon" title="c12">L</font> =  <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">od0</font>,<font color="Maroon" title="c9">om0</font>,<font color="Maroon" title="c7">nm</font>,<font color="Maroon" title="c5">nd</font> #);<br/>
<a NAME="E1:24_1_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">od0</font>,<font color="Maroon" title="c9">om0</font>,<font color="Maroon" title="c7">nm</font>,<font color="Maroon" title="c5">nd</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nd</font>
 
;<br/>
<a NAME="E2:24_1_1"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">od0</font>,<font color="Maroon" title="c9">om0</font>,<font color="Maroon" title="c7">nm</font>,<font color="Maroon" title="c5">nd</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">nm</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">L</font> =  <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c8">od0</font>,<font color="Maroon" title="c9">om0</font>,<font color="Maroon" title="c7">nm</font>,<font color="Maroon" title="c5">nd</font> #) as   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="realset2.html#E5"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a>, <a class="txt" href="realset2.html#E6"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a>, <a class="ref" href="struct_0.html#D8" title="STRUCT_0:def.8">STRUCT_0:def 8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">L</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">L</font> is  <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="realset2.html#V1" title="REALSET2:attr.1">Field-like</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E4:24_1_1"/>
<font color="Maroon" title="c13">L</font> is  <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">L</font> is  <a href="realset2.html#V1" title="REALSET2:attr.1">Field-like</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E5:24_1_1"/>
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c13">L</font> is    <a href="realset1.html#M3" title="REALSET1:mode.3">DnT</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c13">L</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="realset2.html#E39"><span class="lab"><font color="Green" title="E29">Lm28</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="realset2.html#D4" target="_self" title="REALSET2:def.4">REALSET2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c13">L</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default"><a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <font color="Maroon" title="c13">L</font></span>)</span> holds <br/> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</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="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> ) &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )</span><br/>

<a NAME="E6:24_1_1"/>
<font color="Maroon" title="c13">L</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">L</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c14">a</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">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> <font color="Maroon" title="c13">L</font></span><br/>

<a NAME="E1:24_1_1_1"/>
( <font color="Maroon" title="c14">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> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,2) or <font color="Maroon" title="c14">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) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="realset2.html#E5"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a>, <a class="txt" href="realset2.html#E6"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:24_1_1_1"/>
<font color="Maroon" title="c14">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">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> <font color="Maroon" title="c13">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="realset2.html#E12"><span class="lab"><font color="Green" title="E8">Lm8</font></span></a>, <a class="txt" href="realset2.html#E15"><span class="lab"><font color="Green" title="E11">Lm11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:24_1_1"/>
( (  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>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c13">L</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c13">L</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default"><a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <font color="Maroon" title="c13">L</font></span>)</span> holds <br/> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</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="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> ) &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp; <font color="Maroon" title="c13">L</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="realset2.html#E31"><span class="lab"><font color="Green" title="E23">Lm23</font></span></a>, <a class="txt" href="realset2.html#E32"><span class="lab"><font color="Green" title="E24">Lm24</font></span></a>, <a class="txt" href="realset2.html#E33"><span class="lab"><font color="Green" title="E25">Lm25</font></span></a>, <a class="txt" href="realset2.html#E35"><span class="lab"><font color="Green" title="E26">Lm26</font></span></a>, <a class="txt" href="realset2.html#E36"><span class="lab"><font color="Green" title="E27">Lm27</font></span></a>, <a class="ref" href="rlvect_1.html#D2" title="RLVECT_1:def.2">RLVECT_1:def 2</a>, <a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a>, <a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a>, <a class="ref" href="vectsp_1.html#D7" title="VECTSP_1:def.7">VECTSP_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
