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<b>let </b><font color="Maroon" title="c1">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">R</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:64"/><i><font color="Green" title="E50">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="o_ring_1.html#NR2" title="O_RING_1:NR.2">is_a_Sum_of_squares</a> 
 <b>and </b><br/><a NAME="E2:64"/><i><font color="Green" title="E51">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="E3:64"/><i><font color="Green" title="E52">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="nat_1.html#NM1" title="NAT_1:NM.1">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="E4:64"/><i><font color="Green" title="E53">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E5:64"/><i><font color="Green" title="E54">A5</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E55">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="E2:64_1"/>
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>and </b><br/><a NAME="E3:64_1"/><i><font color="Green" title="E56">A7</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="E4:64_1"/><i><font color="Green" title="E57">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#E3:64_1"><i><font color="Green" title="E56">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="E58">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:64_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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="o_ring_1.html#NR1" title="O_RING_1:NR.1">is_a_square</a> 
 <b>by </b><i><a class="txt" href="o_ring_1.html#E1:64"><i><font color="Green" title="E50">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>hence </b><a NAME="E3:64_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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#E46"><i><font color="Green" title="E43">Lm31</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:64_1_2"/><i><font color="Green" title="E59">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="E2:64_1_2"/><b>then </b>
 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="c2">R</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="o_ring_1.html#NR1" title="O_RING_1:NR.1">is_a_square</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="o_ring_1.html#E1:64"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="o_ring_1.html#E4:64_1"><i><font color="Green" title="E57">A8</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>hence </b><a NAME="E3:64_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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#E1:64_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="txt" href="o_ring_1.html#E3:64_1"><i><font color="Green" title="E56">A7</font></i></a>, <a class="txt" href="o_ring_1.html#E1:64_1_2"><i><font color="Green" title="E59">A10</font></i></a>, <a class="txt" href="o_ring_1.html#E51"><i><font color="Green" title="E48">Lm36</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="E7:64_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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#E5:64_1"><i><font color="Green" title="E58">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:64"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="o_ring_1.html#E4:64"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="o_ring_1.html#E5:64"><i><font color="Green" title="E54">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:64"/>
<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#E2:64"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="o_ring_1.html#E3:64"><i><font color="Green" title="E52">A3</font></i></a></i>;<br/>


</div>
