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<span class="kw">let </span><font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font></span>)</span>; <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">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b4">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> ) &amp; ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b4">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> ) implies <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_1_2"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b4">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 <span class="kw">and </span><br/><a NAME="E2:64_1_2"/><span class="lab"><font color="Green" title="E48">A10</font></span>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b4">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><br/><a NAME="E1:64_1_2_1"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">a1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:64_1_2_1"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ;<br/><a NAME="E4:64_1_2_1"/>
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K6" title="LPSPACE2:func.6">CosetSet</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">b1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E6:64_1_2_1"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font> )
 ;<br/><a NAME="E7:64_1_2_1"/><span class="lab"><font color="Green" title="E51">A13</font></span>: 
( <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_2_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E6:64_1_2_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="lpspace2.html#T38" target="_self" title="LPSPACE2:th.38">EQC01</a></span>;<br/><a NAME="E8:64_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lpspace2.html#K5" title="LPSPACE2:func.5">a.e-eq-class_Lp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c10">b1</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_2"><span class="lab"><font color="Green" title="E47">A9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:64_1_2_1"/>
<font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_2"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E7:64_1_2_1"><span class="lab"><font color="Green" title="E51">A13</font></span></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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:64_1_2"/>
<font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</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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