<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="o_ring_1.html#NR2" title="O_RING_1:NR.2">is_a_Sum_of_squares</a> 
 ;<br/>

<a NAME="E2:29"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="o_ring_1.html#NR1" title="O_RING_1:NR.1">is_a_square</a>  )
 
<b>by </b><i><a class="ref" href="o_ring_1.html#D4" target="_self" title="O_RING_1:def.4">Def4</a></i>;<br/>
<a NAME="E3:29"/><b>then </b><i><font color="Green" title="E22">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="o_ring_1.html#NR5" title="O_RING_1:NR.5">is_a_product_of_squares</a>  )
 
<b>by </b><i><a class="txt" href="o_ring_1.html#E16"><i><font color="Green" title="E13">Lm9</font></i></a></i>;<br/>
<a NAME="E4:29"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> holds <br/> ex <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b2">y</font> <a href="o_ring_1.html#NR5" title="O_RING_1:NR.5">is_a_product_of_squares</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:29_1"/>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">y</font> being   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="o_ring_1.html#NR1" title="O_RING_1:NR.1">is_a_square</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="txt" href="o_ring_1.html#E1:29"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="o_ring_1.html#D4" target="_self" title="O_RING_1:def.4">Def4</a></i>;<br/>
<b>thus </b><a NAME="E4:29_1"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="o_ring_1.html#NR5" title="O_RING_1:NR.5">is_a_product_of_squares</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="o_ring_1.html#E3:29_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="o_ring_1.html#E16"><i><font color="Green" title="E13">Lm9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:29"/>
<font color="Maroon" title="c2">f</font> <a href="o_ring_1.html#NR6" title="O_RING_1:NR.6">is_a_Sum_of_products_of_squares</a> 
 <b>by </b><i><a class="txt" href="o_ring_1.html#E3:29"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="o_ring_1.html#D8" target="_self" title="O_RING_1:def.8">Def8</a></i>;<br/>


</div>
