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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">X</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">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b3">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b2">S</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b6">x</font>, <font color="Olive" title="b7">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspace1.html#K20" title="LPSPACE1:func.20">L-1-Space</a> <font color="Olive" title="b3">M</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">x</font> &amp; <font color="Olive" title="b5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b7">y</font> implies <font color="Olive" title="b4">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b7">y</font> ) &amp; ( <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">x</font> implies <font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b6">x</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">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:()"><i><font color="Red">::  thesis: </font></i></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="c2">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><br/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b5">x</font>, <font color="Olive" title="b6">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspace1.html#K20" title="LPSPACE1:func.20">L-1-Space</a> <font color="Olive" title="b2">M</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">x</font> &amp; <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">y</font> implies <font color="Olive" title="b3">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b6">y</font> ) &amp; ( <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">x</font> implies <font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b5">x</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X</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">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S</font><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspace1.html#K20" title="LPSPACE1:func.20">L-1-Space</a> <font color="Olive" title="b1">M</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">x</font> &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">y</font> implies <font color="Olive" title="b2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b5">y</font> ) &amp; ( <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">x</font> implies <font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b4">x</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S</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">f</font>, <font color="Olive" title="b2">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspace1.html#K20" title="LPSPACE1:func.20">L-1-Space</a> <font color="Maroon" title="c4">M</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">y</font> implies <font color="Olive" title="b1">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b4">y</font> ) &amp; ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">x</font> implies <font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspace1.html#K20" title="LPSPACE1:func.20">L-1-Space</a> <font color="Maroon" title="c4">M</font></span>)</span> holds <br/> ( ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">y</font> implies <font color="Maroon" title="c5">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> ) &amp; ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> implies <font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> ) )</span><br/><b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lpspace1.html#K20" title="LPSPACE1:func.20">L-1-Space</a> <font color="Maroon" title="c4">M</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y</font> implies <font color="Maroon" title="c5">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font> ) &amp; ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font> implies <font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font> ) )</span><br/>















<b>set </b><font color="Maroon" title="c9">C</font> =  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c4">M</font>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font> implies <font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:86_1"/><i><font color="Green" title="E58">A2</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span><br/><b>reconsider </b><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c11">y1</font> = <font color="Maroon" title="c8">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="c4">M</font></span>)</span> ;<br/><a NAME="E3:86_1"/>
<font color="Maroon" title="c10">x1</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 <span class="p1">(<span class="default"><a href="lpspace1.html#K18" title="LPSPACE1:func.18">Pre-L-Space</a> <font color="Maroon" title="c4">M</font></span>)</span>
 ;<br/><a NAME="E4:86_1"/><b>then </b><i><font color="Green" title="E59">A3</font></i>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#D18" target="_self" title="LPSPACE1:def.18">VSPDef6X</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">a</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E6:86_1"/><i><font color="Green" title="E60">A4</font></i>: 
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c4">M</font> )
 ;<br/><a NAME="E7:86_1"/><i><font color="Green" title="E61">A5</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x1</font>
 <b>by </b><i><a class="txt" href="#E6:86_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a></i>;<br/><a NAME="E8:86_1"/>
<font color="Maroon" title="c11">y1</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 <span class="p1">(<span class="default"><a href="lpspace1.html#K18" title="LPSPACE1:func.18">Pre-L-Space</a> <font color="Maroon" title="c4">M</font></span>)</span>
 ;<br/><a NAME="E9:86_1"/><b>then </b><i><font color="Green" title="E62">A6</font></i>: 
<font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#D18" target="_self" title="LPSPACE1:def.18">VSPDef6X</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">b</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E11:86_1"/><i><font color="Green" title="E63">A7</font></i>: 
( <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c4">M</font> )
 ;<br/><a NAME="E12:86_1"/>
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">y1</font>
 <b>by </b><i><a class="txt" href="#E11:86_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a></i>;<br/><a NAME="E13:86_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lpspace1.html#K15" title="LPSPACE1:func.15">addCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c13">b</font></span>)</span>,<font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="#E4:86_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="#E9:86_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="#E7:86_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="lpspace1.html#D15" target="_self" title="LPSPACE1:def.15">VSPDef3X</a></i>;<br/><a NAME="E14:86_1"/><b>then </b><i><font color="Green" title="E64">A9</font></i>: 
<font color="Maroon" title="c10">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c13">b</font></span>)</span>,<font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#D18" target="_self" title="LPSPACE1:def.18">VSPDef6X</a></i>;<br/><a NAME="E15:86_1"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c5">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c12">a</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c4">M</font> )
 <b>by </b><i><a class="txt" href="#E1:86_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="#E6:86_1"><i><font color="Green" title="E60">A4</font></i></a></i>;<br/><a NAME="E16:86_1"/><b>then </b><i><font color="Green" title="E65">A11</font></i>: 
 <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#T39" target="_self" title="LPSPACE1:th.39">EQC02</a></i>;<br/><a NAME="E17:86_1"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c6">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c13">b</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c4">M</font> )
 <b>by </b><i><a class="txt" href="#E1:86_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="#E11:86_1"><i><font color="Green" title="E63">A7</font></i></a></i>;<br/><a NAME="E18:86_1"/><b>then </b>
 <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#T39" target="_self" title="LPSPACE1:th.39">EQC02</a></i>;<br/><a NAME="E19:86_1"/><b>then </b>
 <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c13">b</font></span>)</span>,<font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="#E1:86_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="#E6:86_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="#E11:86_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="#E16:86_1"><i><font color="Green" title="E65">A11</font></i></a>, <a class="ref" href="lpspace1.html#T41" target="_self" title="LPSPACE1:th.41">EQC03</a></i>;<br/><b>hence </b><a NAME="E20:86_1"/>
<font color="Maroon" title="c5">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a>, <a class="txt" href="#E14:86_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="lpspace1.html#T23" target="_self" title="LPSPACE1:th.23">Th01a</a>, <a class="txt" href="#E6:86_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="#E1:86_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="#E11:86_1"><i><font color="Green" title="E63">A7</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>
<b>end;</b></div>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span>
<div class="add"><b>assume </b><a NAME="E1:86_2"/><i><font color="Green" title="E58">A15</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font></span><br/><b>reconsider </b><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c7">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="c4">M</font></span>)</span> ;<br/><a NAME="E3:86_2"/>
<font color="Maroon" title="c10">x1</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 <span class="p1">(<span class="default"><a href="lpspace1.html#K18" title="LPSPACE1:func.18">Pre-L-Space</a> <font color="Maroon" title="c4">M</font></span>)</span>
 ;<br/><a NAME="E4:86_2"/><b>then </b><i><font color="Green" title="E59">A16</font></i>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#D18" target="_self" title="LPSPACE1:def.18">VSPDef6X</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E6:86_2"/><i><font color="Green" title="E60">A17</font></i>: 
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c11">f1</font>,<font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c4">M</font> )
 ;<br/><a NAME="E7:86_2"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x1</font>
 <b>by </b><i><a class="txt" href="#E6:86_2"><i><font color="Green" title="E60">A17</font></i></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a></i>;<br/><a NAME="E8:86_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="lpspace1.html#K17" title="LPSPACE1:func.17">lmultCoset</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c11">f1</font></span>)</span>,<font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="#E4:86_2"><i><font color="Green" title="E59">A16</font></i></a>, <a class="ref" href="lpspace1.html#D17" target="_self" title="LPSPACE1:def.17">VSPDef5X</a></i>;<br/><a NAME="E9:86_2"/><b>then </b><i><font color="Green" title="E61">A19</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c11">f1</font></span>)</span>,<font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#D18" target="_self" title="LPSPACE1:def.18">VSPDef6X</a></i>;<br/><a NAME="E10:86_2"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c5">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace1.html#K9" title="LPSPACE1:func.9">L1_Functions</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c4">M</font> )
 <b>by </b><i><a class="txt" href="#E1:86_2"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="#E6:86_2"><i><font color="Green" title="E60">A17</font></i></a></i>;<br/><a NAME="E11:86_2"/><b>then </b>
 <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c11">f1</font>,<font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="ref" href="lpspace1.html#T39" target="_self" title="LPSPACE1:th.39">EQC02</a></i>;<br/><a NAME="E12:86_2"/><b>then </b>
 <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c11">f1</font></span>)</span>,<font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="#E6:86_2"><i><font color="Green" title="E60">A17</font></i></a>, <a class="txt" href="#E1:86_2"><i><font color="Green" title="E58">A15</font></i></a>, <a class="ref" href="lpspace1.html#T42" target="_self" title="LPSPACE1:th.42">EQC04</a></i>;<br/><b>hence </b><a NAME="E13:86_2"/>
<font color="Maroon" title="c1">a</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="#E9:86_2"><i><font color="Green" title="E61">A19</font></i></a>, <a class="ref" href="lpspace1.html#T24" target="_self" title="LPSPACE1:th.24">Th01b</a>, <a class="txt" href="#E1:86_2"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="#E6:86_2"><i><font color="Green" title="E60">A17</font></i></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</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>


</div>
