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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> ;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:71_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K4" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default">2 <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rvsum_1.html#K4" target="_self">+</a> <span class="p2">(<span class="default">2 <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K4" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K4" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="rvsum_1.html#T78" target="_self">Th78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="ref" href="rvsum_1.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T78" target="_self">Th78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T78" target="_self">Th78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T87" target="_self">Th87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66" target="_self">Th66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rvsum_1.html#K4" target="_self">+</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rvsum_1.html#K4" target="_self">+</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K4" target="_self">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T27" target="_self">Th27</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:71"/>
 <a href="rvsum_1.html#K12" target="_self">sqr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K4" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rvsum_1.html#K4" target="_self">+</a> <span class="p2">(<span class="default">2 <a href="rvsum_1.html#K10" target="_self">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14" target="_self">mlt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K4" target="_self">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12" target="_self">sqr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>;<br/>


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