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<b>set </b><font color="Maroon" title="c4">C</font> =  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>let </b><font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<span class="p2">(<span class="default"><a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</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">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font><br/> for <font color="Olive" title="b3">f</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">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font> holds <br/><font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</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="Olive" title="b1">z</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Maroon" title="c3">M</font> ) &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font><br/> for <font color="Olive" title="b3">f</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">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font> holds <br/><font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</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="Olive" title="b1">z</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Maroon" title="c3">M</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/>

<b>assume </b><b>that </b><br/><a NAME="E1:73_1_2"/><i><font color="Green" title="E45">A8</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font><br/> for <font color="Olive" title="b3">a</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="b2">A</font> holds <br/><font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</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="Olive" title="b1">z</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b3">a</font></span>)</span>,<font color="Maroon" title="c3">M</font>
 <b>and </b><br/><a NAME="E2:73_1_2"/><i><font color="Green" title="E46">A9</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font><br/> for <font color="Olive" title="b3">a</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="b2">A</font> holds <br/><font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">A</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="Olive" title="b1">z</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b3">a</font></span>)</span>,<font color="Maroon" title="c3">M</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">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="73_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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">A</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</font> holds  <font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">z</font>,<font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">z</font>,<font color="Olive" title="b1">A</font></span><br/><b>let </b><font color="Maroon" title="c8">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="lpspace1.html#K14" title="LPSPACE1:func.14">CosetSet</a> <font color="Maroon" title="c3">M</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">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">A</font></span><br/><a NAME="E1:73_1_2_1"/>
<font color="Maroon" title="c8">A</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="c3">M</font>
 ;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:73_1_2_1"/><i><font color="Green" title="E47">A10</font></i>: 
( <font color="Maroon" title="c8">A</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="c9">a1</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">a1</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="c3">M</font> )
 ;<br/><b>thus </b><font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">A</font> = 
 <a href="lpspace1.html#K13" title="LPSPACE1:func.13">a.e-eq-class</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c9">a1</font></span>)</span>,<font color="Maroon" title="c3">M</font>
<b>by </b><i><a class="txt" href="#E1:73_1_2"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="#E3:73_1_2_1"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="lpspace1.html#T38" target="_self" title="LPSPACE1:th.38">EQC01</a></i>
<br/>.= 
<font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">A</font>
<b>by </b><i><a class="txt" href="#E2:73_1_2"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="#E3:73_1_2_1"><i><font color="Green" title="E47">A10</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>
<b>hence </b><a NAME="E4:73_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>
 <b>by </b><i><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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