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<span class="kw">set </span><font color="Maroon" title="c5">C</font> =  <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">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">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="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 holds <br/>$3 <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="b1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>);<br/>
<div><span class="lab"><font color="Green" title="E47">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c7">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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</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>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font>,<font color="Olive" title="b1">z</font>]</span><br/><a NAME="E1:64_1_1_1"/>
<font color="Maroon" title="c6">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="c8">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><span class="kw"> such that </span><br/><a NAME="E3:64_1_1_1"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
( <font color="Maroon" title="c6">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="c8">a</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) &amp; <font color="Maroon" title="c8">a</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_1_1"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
 ex <font color="Olive" title="b1">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp; <font color="Maroon" title="c8">a</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="lpspace2.html#T35" target="_self" title="LPSPACE2:th.35">Th35</a></span>;<br/><a NAME="E5:64_1_1_1"/>
<font color="Maroon" title="c7">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="c9">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><span class="kw"> such that </span><br/><a NAME="E7:64_1_1_1"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
( <font color="Maroon" title="c7">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="c9">b</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) &amp; <font color="Maroon" title="c9">b</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="E8:64_1_1_1"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
 ex <font color="Olive" title="b1">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp; <font color="Maroon" title="c9">b</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64_1_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="lpspace2.html#T35" target="_self" title="LPSPACE2:th.35">Th35</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">z</font> =  <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="c8">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>);<br/><a NAME="E9:64_1_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c9">b</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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace2.html#T25" target="_self" title="LPSPACE2:th.25">Th25</a>, <a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E7:64_1_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a></span>;<br/><a NAME="E10:64_1_1_1"/><span class="kw">then </span>
 <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="c8">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</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">reconsider </span><font color="Maroon" title="c11">z</font> =  <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="c8">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) as    <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/><span class="kw">take </span><font color="Maroon" title="c12">z</font> = <font color="Maroon" title="c11">z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c12">z</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">a1</font>, <font color="Maroon" title="c14">b1</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c14">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> implies <font color="Maroon" title="c12">z</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="c13">a1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">b1</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_1_1_1"/>
( <font color="Maroon" title="c13">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c14">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">z</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="c13">a1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">b1</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span><br/><a NAME="E2:64_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">a1</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c14">b1</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c3">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E4:64_1_1_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E7:64_1_1_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E8:64_1_1_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="lpspace2.html#T37" target="_self" title="LPSPACE2:th.37">Th37</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_1_1"/>
<font color="Maroon" title="c12">z</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="c13">a1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c14">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="ref" href="lpspace2.html#T42" target="_self" title="LPSPACE2:th.42">Th42</a>, <a class="ref" href="lpspace1.html#T31" title="LPSPACE1:th.31">LPSPACE1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:64_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c12">z</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <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"><span class="p2">(<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>,<span class="p2">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<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><span class="kw"> such that </span><br/><a NAME="E3:64_1_1"/><span class="lab"><font color="Green" title="E48">A6</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>) holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Maroon" title="c6">f</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>)]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E47">A1</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as   <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> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c7">f</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><br/>

<span class="kw">let </span><font color="Maroon" title="c8">A</font>, <font color="Maroon" title="c9">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="firebrick">::  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="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="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> holds <br/><font color="Maroon" title="c7">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">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="b1">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> implies <font color="Maroon" title="c7">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">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="c10">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c11">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E5:64_1_1"/>
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">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="c10">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c11">b</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">k</font>)</span><br/>

<span class="kw">hence </span><a NAME="E6:64_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">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="c10">a</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c11">b</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="#E3:64_1_1"><span class="lab"><font color="Green" title="E48">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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