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<span class="kw">let </span><font color="Maroon" title="c1">D</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">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">X</font></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  holds <br/> <a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b1">F</font></span>)</span>,<font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">X</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">D</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">X</font></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  holds <br/> <a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span>,<font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <font color="Maroon" title="c2">F</font>,<font color="Olive" title="b1">X</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be    <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">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  holds <br/> <a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">X</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  implies  <a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">X</font></span>)</span> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">x</font> =  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:105"/><span class="lab"><font color="Green" title="E65">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">X</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">FX</font> = <font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="finset_1.html#T29" title="FINSET_1:th.29">FINSET_1:29</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T65" title="RFUNCT_1:th.65">RFUNCT_1:65</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">rFX</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="ref" href="finset_1.html#T29" title="FINSET_1:th.29">FINSET_1:29</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">b</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E6:105"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span>,<font color="Maroon" title="c8">b</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="ref" href="rfinseq.html#T18" title="RFINSEQ:th.18">RFINSEQ:18</a></span>;<br/>
<a NAME="E7:105"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:105"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="classes1.html#T83" title="CLASSES1:th.83">CLASSES1:83</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c8">b</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E9:105"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E67">A3</font></span>: <font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></span>
<br/>.= 
<font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></span>
;<br/>
<a NAME="E11:105"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A4</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:105"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="classes1.html#T83" title="CLASSES1:th.83">CLASSES1:83</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E69">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <font color="Maroon" title="c7">rFX</font>,<font color="Maroon" title="c10">x</font></span>)</span></span><br/><a NAME="E1:105_3"/><span class="lab"><font color="Green" title="E70">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T37" title="FINSEQ_2:th.37">FINSEQ_2:37</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:105_3_1_1"/>
(  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:105_3_1_1_1"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rFX</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E72">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:105_3_1_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:105_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T65" title="RFUNCT_1:th.65">RFUNCT_1:65</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><span class="kw"> such that </span><br/><a NAME="E4:105_3_1_1_1_1"/><span class="lab"><font color="Green" title="E73">A9</font></span>: 
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:105_3_1_1_1_1"/><span class="lab"><font color="Green" title="E74">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></span>;<br/><a NAME="E6:105_3_1_1_1_1"/>
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105_3_1_1_1_1"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/><a NAME="E7:105_3_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:105_3_1_1_1_1"/><span class="lab"><font color="Green" title="E75">A11</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">and </span><br/><a NAME="E10:105_3_1_1_1_1"/><span class="lab"><font color="Green" title="E76">A12</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:105"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><a NAME="E11:105_3_1_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105_3_1_1_1_1"><span class="lab"><font color="Green" title="E73">A9</font></span></a>, <a class="txt" href="#E5:105_3_1_1_1_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/><a NAME="E12:105_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A13</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:105_3_1_1_1_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></span>;<br/><a NAME="E13:105_3_1_1_1_1"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_3"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="txt" href="#E9:105_3_1_1_1_1"><span class="lab"><font color="Green" title="E75">A11</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><span class="kw">hence </span><a NAME="E14:105_3_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_3_1_1_1"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E12:105_3_1_1_1_1"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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><a NAME="E3:105_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_3_1_1_1"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="rfunct_3.html#E9"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:105_3_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rFX</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105_3_1_1_1"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="txt" href="rfunct_3.html#E9"><span class="lab"><font color="Green" title="E7">Lm2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:105_3_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rFX</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:105_3_1_1_2"/>
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:105_3_1_1_2"/><span class="lab"><font color="Green" title="E71">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c9">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E6:105_3_1_1_2"/><span class="lab"><font color="Green" title="E72">A15</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:105_3_1_1_2"><span class="lab"><font color="Green" title="E71">A14</font></span></a>, <a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:105_3_1_1_2_1_1"/>
( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:105_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E73">A16</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rFX</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:105_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:105_3_1_1_2"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="ref" href="rfinseq.html#T38" title="RFINSEQ:th.38">RFINSEQ:38</a></span>;<br/> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">FX</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:105_3_1_1_2"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="txt" href="#E1:105_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="ref" href="rfunct_3.html#T9" target="_self" title="RFUNCT_3:th.9">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T65" title="RFUNCT_1:th.65">RFUNCT_1:65</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:105_3_1_1_2_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rFX</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:105"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E10:105"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E2:105_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A17</font></span></a>, <a class="ref" href="classes1.html#T89" title="CLASSES1:th.89">CLASSES1:89</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:105_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E73">A18</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <font color="Maroon" title="c7">rFX</font>,<font color="Maroon" title="c10">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:105_3_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A19</font></span>: 
 <a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <font color="Maroon" title="c9">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T37" title="RFINSEQ:th.37">RFINSEQ:37</a></span>;<br/> <a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>,<font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T65" title="RFUNCT_1:th.65">RFUNCT_1:65</a></span>
<br/>.= 
 <a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <font color="Maroon" title="c6">FX</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A18</font></span></a>, <a class="ref" href="rfunct_3.html#T8" target="_self" title="RFUNCT_3:th.8">Th8</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:105_3_1_1_2_1_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <font color="Maroon" title="c7">rFX</font>,<font color="Maroon" title="c10">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:105"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E10:105"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E2:105_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A19</font></span></a>, <a class="ref" href="classes1.html#D9" title="CLASSES1:def.9">CLASSES1:def 9</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:105_3_1_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">rFX</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:105_3"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K12" title="RELAT_1:func.12">Coim</a> <font color="Maroon" title="c7">rFX</font>,<font color="Maroon" title="c10">x</font></span>)</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>
<a NAME="E13:105"/>
(  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
;<br/>
<a NAME="E14:105"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A20</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:105"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="classes1.html#T87" title="CLASSES1:th.87">CLASSES1:87</a></span>;<br/>
<a NAME="E15:105"/>
<font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font>, <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">X</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="ref" href="rfunct_3.html#D14" target="_self" title="RFUNCT_3:def.14">Def14</a></span>;<br/>
<a NAME="E16:105"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A21</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:105"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E10:105"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="classes1.html#T84" title="CLASSES1:th.84">CLASSES1:84</a>, <a class="ref" href="rfinseq.html#T22" title="RFINSEQ:th.22">RFINSEQ:22</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c3">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
;<br/>
<a NAME="E18:105"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font>, <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span>,<font color="Maroon" title="c3">X</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="ref" href="rfunct_3.html#D14" target="_self" title="RFUNCT_3:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span> <a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">F</font></span>)</span>,<font color="Maroon" title="c3">X</font> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c9">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:105"><span class="lab"><font color="Green" title="E70">A20</font></span></a>, <a class="ref" href="classes1.html#T84" title="CLASSES1:th.84">CLASSES1:84</a>, <a class="ref" href="rfinseq.html#T22" title="RFINSEQ:th.22">RFINSEQ:22</a></span>
<br/>.= 
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:105"><span class="lab"><font color="Green" title="E71">A21</font></span></a>, <a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></span>
;<br/> <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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