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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font>;<br/>



<b>assume </b><a NAME="E1:48"/>
<font color="Maroon">x</font> <a href="o_ring_1.html#NR5" target="_self">is_a_product_of_squares</a> 
 ;<br/>

<a NAME="E2:48"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="o_ring_1.html#NR9" target="_self">is_an_amalgam_of_squares</a> 
 
<b>by </b><i><a class="txt" href="o_ring_1.html#E3"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E3:48"/><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E4:48"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="o_ring_1.html#NR9" target="_self">is_an_amalgam_of_squares</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:48"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> holds <br/> ex <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">y</font> <a href="o_ring_1.html#NR9" target="_self">is_an_amalgam_of_squares</a>  &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:48_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/>

<a NAME="E2:48_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:48_1"/>
 ex <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">y</font> <a href="o_ring_1.html#NR9" target="_self">is_an_amalgam_of_squares</a>  &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:48"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="o_ring_1.html#NR10" target="_self">is_a_Sum_of_amalgams_of_squares</a> 
 <b>by </b><i>, , <a class="txt" href="o_ring_1.html#E2:48"><i><font color="Green">E29</font></i></a></i>;<br/>


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