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<div class="add">

<b>let </b><font color="Maroon" title="c12">L</font> be  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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> implies <font color="Maroon" title="c12">L</font> is <a href="realset2.html#V1" title="REALSET2:attr.1">Field-like</a> )</span><br/>

<b>assume </b><a NAME="E1:26_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> &amp; <font color="Maroon" title="c12">L</font> is <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">L</font> is <a href="realset2.html#V1" title="REALSET2:attr.1">Field-like</a></span><br/>

<b>set </b><font color="Maroon" title="c13">B</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E2:26_1_1"/><i><font color="Green" title="E28">A3</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</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> holds <br/>the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c12">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">z</font> <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</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> implies the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c12">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:26_1_1_1"/>
<font color="Maroon" title="c14">z</font> <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c12">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:26_1_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>and </b><br/><a NAME="E4:26_1_1_1"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c16">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">L</font> <b>by </b><i><a class="txt" href="#E3:26_1_1_1"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/>
<a NAME="E6:26_1_1_1"/>
( not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="#E3:26_1_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E7:26_1_1_1"/><b>then </b>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c12">L</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c12">L</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:26_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c12">L</font>
 
<b>by </b><i><a class="txt" href="#E1:26_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/>
<a NAME="E9:26_1_1_1"/><b>then </b>
not <font color="Maroon" title="c17">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c12">L</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:26_1_1_1"/>
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c12">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:26_1_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_1_1"/>
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c12">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="c12">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</font>
 <b>by </b><i><a class="ref" href="realset1.html#D8" title="REALSET1:def.8">REALSET1:def 8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="realset2.html#D11" target="_self" title="REALSET2:def.11">REALSET2:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &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="c12">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="c12">L</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></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></span><br/>

<b>reconsider </b><font color="Maroon" title="c14">om</font> = the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c12">L</font> as    <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="c12">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</font> <b>by </b><i><a class="txt" href="#E2:26_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="realset1.html#D8" title="REALSET1:def.8">REALSET1:def 8</a></i>;<br/>
<b>let </b><font color="Maroon" title="c15">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c15">B</font><br/> for <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">B</font>  st <font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b2">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="c12">L</font> &amp; <font color="Olive" title="b1">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="c12">L</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> holds <br/> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">e</font> #) is   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a></span><br/><b>let </b><font color="Maroon" title="c16">P</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c15">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">B</font>  st <font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b1">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="c12">L</font> &amp; <font color="Maroon" title="c16">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="c12">L</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> holds <br/> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Olive" title="b1">e</font> #) is   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a></span><br/><b>let </b><font color="Maroon" title="c17">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c17">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="c12">L</font> &amp; <font color="Maroon" title="c16">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="c12">L</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> implies  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #) is   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E5:26_1_1"/><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E6:26_1_1"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c17">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="c12">L</font>
 <b>and </b><br/><a NAME="E7:26_1_1"/><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c16">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="c12">L</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">L</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #) is   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a></span><br/>

<b>set </b><font color="Maroon" title="c18">K</font> =  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #);<br/>
<a NAME="E8:26_1_1"/><i><font color="Green" title="E32">A9</font></i>: 
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #) is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">v</font>, <font color="Maroon" title="c20">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c20">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c19">v</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c21">a</font> = <font color="Maroon" title="c19">v</font>, <font color="Maroon" title="c22">b</font> = <font color="Maroon" title="c20">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">L</font> <b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E2:26_1_1_2"/><i><font color="Green" title="E33">A10</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c20">w</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"><font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c15">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<a NAME="E3:26_1_1_2"/><i><font color="Green" title="E34">A11</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">w</font>,<font color="Maroon" title="c19">v</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"><font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c15">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<b>thus </b><font color="Maroon" title="c19">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c20">w</font> = 
<font color="Maroon" title="c21">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c22">b</font>
<b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E7:26_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="#E2:26_1_1_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c22">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">a</font>
<b>by </b><i><a class="txt" href="#E1:26_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a></i>
<br/>.= 
<font color="Maroon" title="c20">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c19">v</font>
<b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E7:26_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="#E3:26_1_1_2"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:26_1_1"/><i><font color="Green" title="E33">A12</font></i>: 
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">u</font>, <font color="Maroon" title="c20">v</font>, <font color="Maroon" title="c21">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c19">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c21">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c21">w</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c22">a</font> = <font color="Maroon" title="c19">u</font>, <font color="Maroon" title="c23">b</font> = <font color="Maroon" title="c20">v</font>, <font color="Maroon" title="c24">c</font> = <font color="Maroon" title="c21">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">L</font> <b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E2:26_1_1_3"/><i><font color="Green" title="E34">A13</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</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"><font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c15">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<a NAME="E3:26_1_1_3"/><b>then </b>
<font color="Maroon" title="c16">P</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E4:26_1_1_3"/><b>then </b><i><font color="Green" title="E35">A14</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">om</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c15">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font></span>)</span>,<font color="Maroon" title="c21">w</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="c15">B</font>,<font color="Maroon" title="c15">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E7:26_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:26_1_1_3"/><i><font color="Green" title="E36">A15</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c21">w</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"><font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c15">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<a NAME="E6:26_1_1_3"/><b>then </b>
<font color="Maroon" title="c16">P</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c21">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E7:26_1_1_3"/><b>then </b><i><font color="Green" title="E37">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">u</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">om</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c15">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c21">w</font></span>)</span></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="c15">B</font>,<font color="Maroon" title="c15">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E7:26_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c19">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c20">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c21">w</font> = 
<font color="Maroon" title="c14">om</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">om</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c15">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font></span>)</span>,<font color="Maroon" title="c21">w</font>
<b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E7:26_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="#E4:26_1_1_3"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c23">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c24">c</font>
<b>by </b><i><a class="txt" href="#E2:26_1_1_3"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c22">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c24">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:26_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c14">om</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">u</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">om</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c15">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c21">w</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:26_1_1_3"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c19">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c21">w</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E7:26_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="#E7:26_1_1_3"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:26_1_1"/><i><font color="Green" title="E34">A17</font></i>: 
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c20">a</font> = <font color="Maroon" title="c19">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">L</font> <b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E2:26_1_1_4"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #)</span>)</span></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"><font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c15">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<b>hence </b><font color="Maroon" title="c19">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #)</span>)</span> = 
<font color="Maroon" title="c20">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c12">L</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E6:26_1_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="#E7:26_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c19">v</font>
<b>by </b><i><a class="txt" href="#E1:26_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:26_1_1"/>
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</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="26_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #); <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">v</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c20">a</font> = <font color="Maroon" title="c19">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">L</font> <b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E2:26_1_1_5"/>
not <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c12">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E3:26_1_1_5"/><b>then </b>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c12">L</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">L</font><b> such that </b><br/><a NAME="E5:26_1_1_5"/><i><font color="Green" title="E35">A18</font></i>: 
<font color="Maroon" title="c21">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">a</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="c12">L</font>
 <b>by </b><i><a class="txt" href="#E1:26_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="vectsp_1.html#D20" title="VECTSP_1:def.20">VECTSP_1:def 20</a></i>;<br/>
<a NAME="E6:26_1_1_5"/><i><font color="Green" title="E36">A19</font></i>: 
<font color="Maroon" title="c20">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">b</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="c12">L</font>
 
<b>by </b><i><a class="txt" href="#E1:26_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="#E5:26_1_1_5"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a></i>;<br/>
<a NAME="E7:26_1_1_5"/><b>then </b>
<font color="Maroon" title="c21">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c12">L</font>
 
<b>by </b><i><a class="txt" href="#E1:26_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>;<br/>
<a NAME="E8:26_1_1_5"/><b>then </b>
not <font color="Maroon" title="c21">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c12">L</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">w</font> = <font color="Maroon" title="c21">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #) <b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c22">w</font>
; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c22">w</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="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #)</span><br/>

<a NAME="E10:26_1_1_5"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c22">w</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"><font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c15">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<b>hence </b><a NAME="E11:26_1_1_5"/>
<font color="Maroon" title="c19">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c22">w</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="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #)
 <b>by </b><i><a class="txt" href="#E5:26_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="#E6:26_1_1"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="#E7:26_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="#E6:26_1_1_5"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:26_1_1"/>
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <font color="Maroon" title="c15">B</font>,<font color="Maroon" title="c16">P</font>,<font color="Maroon" title="c17">e</font> #) is   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a>
 <b>by </b><i><a class="txt" href="#E8:26_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="#E9:26_1_1"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="#E10:26_1_1"><i><font color="Green" title="E34">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
