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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">R</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:90"/><i><font color="Green">E48</font></i>: 
 <a href="rvsum_1.html#K15" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:90"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:90"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:90"/><b>then </b><i><font color="Green">E61</font></i>: 
(  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">R</font> &amp;  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> )
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E3:90"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E5:90"/>
<font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> <a href="finseqop.html#K4">|-&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:90"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E8:90"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E3:90"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rvsum_1.html#E4:90"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<a NAME="E9:90"/><i><font color="Green">E72</font></i>: 
(  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">R</font></span>)</span></span>)</span> &amp;  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><i><font color="Green">E122</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:90_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">R</font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">R</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>;<br/><b>set </b><font color="Maroon">x</font> = <font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>;<br/><a NAME="E2:90_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><b>hence </b><a NAME="E3:90_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">R</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E11:90"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E7:90"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<b>set </b><font color="Maroon">x</font> = <font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">x'</font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">R</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font>;<br/>
<a NAME="E12:90"/>
<font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E3:90"><i><font color="Green">E60</font></i></a>, <a class="txt" href="rvsum_1.html#E7:90"><i><font color="Green">E63</font></i></a>, , , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E13:90"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/>
<a NAME="E14:90"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">R</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:90"/>
contradiction
 <b>by </b><i><a class="ref" href="rvsum_1.html#T23" target="_self">Th23</a>, <a class="ref" href="rvsum_1.html#T26" target="_self">Th26</a>, <a class="ref" href="rvsum_1.html#T27" target="_self">Th27</a>, </i>;<br/>


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