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<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">b2</font> be   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E6">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon" title="c2">b1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">S</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b2</font></span>)</span>;<br/>
<a NAME="E2:5"/><i><font color="Green" title="E7">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:5"/><b>then </b>
(  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c2">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">S</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b2</font></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E5:5"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c2">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">f1</font> being    <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> <b> such that </b><br/><a NAME="E7:5"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c5">S</font></span>)</span>
 <b>and </b><br/><a NAME="E8:5"/><i><font color="Green" title="E9">A4</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c6">f1</font>
 <b>by </b><i><a class="ref" href="uproots.html#T16" title="UPROOTS:th.16">UPROOTS:16</a></i>;<br/>
<a NAME="E9:5"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">f2</font> being    <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> <b> such that </b><br/><a NAME="E11:5"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c5">S</font></span>)</span>
 <b>and </b><br/><a NAME="E12:5"/><i><font color="Green" title="E11">A6</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c7">f2</font>
 <b>by </b><i><a class="ref" href="uproots.html#T16" title="UPROOTS:th.16">UPROOTS:16</a></i>;<br/>
<a NAME="E13:5"/><i><font color="Green" title="E12">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5" title="UPROOTS:th.5">UPROOTS:5</a></i>;<br/>
<a NAME="E14:5"/><b>then </b><i><font color="Green" title="E13">A8</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c5">S</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c5">S</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="glib_004.html#E2:5"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="glib_004.html#E7:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="glib_004.html#E11:5"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E15:5"/><b>then </b><i><font color="Green" title="E14">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f1</font> <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="c7">f2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E16:5"/><i><font color="Green" title="E15">A10</font></i>: 
<font color="Maroon" title="c6">f1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E17:5"/><i><font color="Green" title="E16">A11</font></i>: 
<font color="Maroon" title="c7">f2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f1</font></span>)</span>
 ;<br/><a NAME="E2:5_1"/><b>then </b><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c8">j</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="c6">f1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:5_1"/><b>then </b><i><font color="Green" title="E18">A13</font></i>: 
( <font color="Maroon" title="c6">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> &amp; <font color="Maroon" title="c7">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E7:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="glib_004.html#E11:5"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="glib_004.html#E14:5"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E4:5_1"/>
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E13:5"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="glib_004.html#E14:5"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="glib_004.html#E2:5_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E5:5_1"/>
<font color="Maroon" title="c6">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="glib_004.html#E3:5_1"><i><font color="Green" title="E18">A13</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:5"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c2">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c3">b2</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E8:5"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="glib_004.html#E12:5"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="glib_004.html#E15:5"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="glib_004.html#E16:5"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="glib_004.html#E17:5"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="rvsum_1.html#T112" title="RVSUM_1:th.112">RVSUM_1:112</a></i>;<br/>


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