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<b>let </b><font color="Maroon" title="c1">fs</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:7"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c1">fs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">fs'</font> = <font color="Maroon" title="c1">fs</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>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:7_1"/><i><font color="Green" title="E5">A1</font></i>: 
 <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Maroon" title="c1">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:7_1"/><i><font color="Green" title="E6">A2</font></i>: 
 <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="c1">fs</font></span>)</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="c1">fs</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="E3:7_1"/><i><font color="Green" title="E7">A3</font></i>: 
 <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="c1">fs</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fs</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">k</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="E1:7_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c3">k</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="c1">fs</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_1_1"/>
<font color="Maroon" title="c1">fs</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:7_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">fs</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c3">k</font>
 ;<br/><a NAME="E3:7_1_1_1"/><b>then </b><i><font color="Green" title="E9">A5</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <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>  st <br/>( <font color="Olive" title="b1">k</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="c2">fs'</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">fs</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E2:7_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="bagorder.html#E1:7_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/><a NAME="E4:7_1_1_1"/>
for <font color="Olive" title="b1">k</font> being   <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>   st <font color="Olive" title="b1">k</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="c1">fs</font> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">fs</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font>
 ;<br/><b>hence </b><a NAME="E5:7_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E1:7_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="bagorder.html#E3:7_1_1_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="rvsum_1.html#T115" title="RVSUM_1:th.115">RVSUM_1:115</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:7_1_1"/>
<font color="Maroon" title="c1">fs</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fs</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0</span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:7_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:7_1"/>
<font color="Maroon" title="c1">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fs</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0
 <b>by </b><i><a class="txt" href="bagorder.html#E2:7_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="bagorder.html#E3:7_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:7"/>
<font color="Maroon" title="c1">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fs</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 0
 ;<br/>

<b>hence </b><a NAME="E5:7"/>
 <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Maroon" title="c1">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T111" title="RVSUM_1:th.111">RVSUM_1:111</a></i>;<br/>


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