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<span class="kw">let </span><font color="Maroon" title="c1">X</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font> holds   <a href="lpspace1.html#K18" title="LPSPACE1:func.18">Pre-L-Space</a> <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Olive" title="b2">M</font>,1</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font> holds   <a href="lpspace1.html#K18" title="LPSPACE1:func.18">Pre-L-Space</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Olive" title="b1">M</font>,1</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="lpspace1.html#K18" title="LPSPACE1:func.18">Pre-L-Space</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,1</span><br/>





<a NAME="E1:110"/><span class="lab"><font color="Green" title="E87">A1</font></span>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lpspace1.html#K18" title="LPSPACE1:func.18">Pre-L-Space</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="lpspace1.html#K18" title="LPSPACE1:func.18">Pre-L-Space</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="lpspace1.html#K18" title="LPSPACE1:func.18">Pre-L-Space</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font> &amp; the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <span class="p1">(<span class="default"><a href="lpspace1.html#K18" title="LPSPACE1:func.18">Pre-L-Space</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#D18" title="LPSPACE1:def.18">LPSPACE1:def 18</a></span>;<br/>
<a NAME="E2:110"/>
(  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,1 &amp;  <a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c3">M</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace2.html#K7" title="LPSPACE2:func.7">addCoset</a> <font color="Maroon" title="c3">M</font>,1 &amp;  <a href="lpspace1.html#K16" title="LPSPACE1:func.16">zeroCoset</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K8" title="LPSPACE2:func.8">zeroCoset</a> <font color="Maroon" title="c3">M</font>,1 &amp;  <a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c3">M</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="lpspace2.html#K9" title="LPSPACE2:func.9">lmultCoset</a> <font color="Maroon" title="c3">M</font>,1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T72" target="_self" title="LPSPACE2:th.72">Th9a</a>, <a class="ref" href="lpspace2.html#T73" target="_self" title="LPSPACE2:th.73">Th9b</a>, <a class="ref" href="lpspace2.html#T74" target="_self" title="LPSPACE2:th.74">Th9c</a>, <a class="ref" href="lpspace2.html#T75" target="_self" title="LPSPACE2:th.75">Th9d</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:110"/>
 <a href="lpspace1.html#K18" title="LPSPACE1:func.18">Pre-L-Space</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K10" title="LPSPACE2:func.10">Pre-Lp-Space</a> <font color="Maroon" title="c3">M</font>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="ref" href="lpspace2.html#D11" target="_self" title="LPSPACE2:def.11">VSPDef6X</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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