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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</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="c2">y</font> implies <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E25">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</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="c2">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span><br/>

<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T12" target="_self" title="EUCLID_2:th.12">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c1">x</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c2">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="euclid_2.html#E1:29"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="euclid_2.html#T31" target="_self" title="EUCLID_2:th.31">Th31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c2">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T12" target="_self" title="EUCLID_2:th.12">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#T12" target="_self" title="EUCLID_2:th.12">Th12</a></i>
;<br/>
<b>hence </b><a NAME="E3:29"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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