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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/>



<a NAME="E1:70"/><i><font color="Green">E48</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="E2:70"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" target="_self">sqrreal</a>  <a href="finseqop.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E3:70"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">R</font>,<font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/>
<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E2:70_1"/><b>then </b><i><font color="Green">E71</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#K1" target="_self">sqrreal</a>  <a href="finseqop.html#K5">*</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/><a NAME="E3:70_1"/><i><font color="Green">E72</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="binop_2.html#K35">multreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">R</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T3" target="_self">Th3</a>, <a class="ref" href="rvsum_1.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">r</font> = <font color="Maroon">R</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="rvsum_1.html#K1" target="_self">sqrreal</a>  <a href="finseqop.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> = 
<a href="rvsum_1.html#K1" target="_self">sqrreal</a>  <a href="seq_1.html#K2">.</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>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<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/>
<br/>.= 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</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>,<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>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">R</font>,<font color="Maroon">R</font></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#E1:70"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:70"/>
 <a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">R</font>,<font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T3" target="_self">Th3</a>, <a class="txt" href="rvsum_1.html#E7"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


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