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<b>let </b><font color="Maroon" title="c1">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">R</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:79"/><i><font color="Green" title="E65">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="o_ring_1.html#NR6" title="O_RING_1:NR.6">is_a_Sum_of_products_of_squares</a> 
 <b>and </b><br/><a NAME="E2:79"/><i><font color="Green" title="E66">A2</font></i>: 
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">i</font>
 <b>and </b><br/><a NAME="E3:79"/><i><font color="Green" title="E67">A3</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <a href="o_ring_1.html#NR13" title="O_RING_1:NR.13">is_generated_from_squares</a>  );<br/>
<a NAME="E4:79"/><i><font color="Green" title="E68">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E5:79"/><i><font color="Green" title="E69">A5</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:79_1"/><i><font color="Green" title="E70">A6</font></i>: 
( 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font> <a href="o_ring_1.html#NR13" title="O_RING_1:NR.13">is_generated_from_squares</a>  )
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:79_1"/>
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>and </b><br/><a NAME="E3:79_1"/><i><font color="Green" title="E71">A7</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<a NAME="E4:79_1"/><i><font color="Green" title="E72">A8</font></i>: 
<font color="Maroon" title="c4">i</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="c3">f</font>
 
<b>by </b><i><a class="txt" href="o_ring_1.html#E3:79_1"><i><font color="Green" title="E71">A7</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><i><font color="Green" title="E73">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:79_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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#E1:79"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="o_ring_1.html#D8" target="_self" title="O_RING_1:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E3:79_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="o_ring_1.html#NR13" title="O_RING_1:NR.13">is_generated_from_squares</a> 
 <b>by </b><i><a class="txt" href="o_ring_1.html#E63"><i><font color="Green" title="E60">Lm48</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_1_2"/><i><font color="Green" title="E74">A10</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:79_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Scalar</a> of <font color="Maroon" title="c2">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="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</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="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="o_ring_1.html#E1:79"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="o_ring_1.html#E4:79_1"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="o_ring_1.html#D8" target="_self" title="O_RING_1:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E3:79_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="o_ring_1.html#NR13" title="O_RING_1:NR.13">is_generated_from_squares</a> 
 <b>by </b><i><a class="txt" href="o_ring_1.html#E1:79_1"><i><font color="Green" title="E70">A6</font></i></a>, <a class="txt" href="o_ring_1.html#E3:79_1"><i><font color="Green" title="E71">A7</font></i></a>, <a class="txt" href="o_ring_1.html#E1:79_1_2"><i><font color="Green" title="E74">A10</font></i></a>, <a class="txt" href="o_ring_1.html#E66"><i><font color="Green" title="E63">Lm51</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:79_1"/>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="o_ring_1.html#NR13" title="O_RING_1:NR.13">is_generated_from_squares</a> 
 <b>by </b><i><a class="txt" href="o_ring_1.html#E5:79_1"><i><font color="Green" title="E73">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:79"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="o_ring_1.html#E4:79"><i><font color="Green" title="E68">A4</font></i></a>, <a class="txt" href="o_ring_1.html#E5:79"><i><font color="Green" title="E69">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:79"/>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">i</font> <a href="o_ring_1.html#NR13" title="O_RING_1:NR.13">is_generated_from_squares</a> 
 <b>by </b><i><a class="txt" href="o_ring_1.html#E2:79"><i><font color="Green" title="E66">A2</font></i></a>, <a class="txt" href="o_ring_1.html#E3:79"><i><font color="Green" title="E67">A3</font></i></a></i>;<br/>


</div>
