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<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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">x</font> is <a href="o_ring_1.html#V7" title="O_RING_1:attr.7">being_a_sum_of_products_of_squares</a> holds <br/><font color="Olive" title="b1">x</font> is <a href="o_ring_1.html#V13" title="O_RING_1:attr.13">generated_from_squares</a></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> is <a href="o_ring_1.html#V7" title="O_RING_1:attr.7">being_a_sum_of_products_of_squares</a> implies <font color="Maroon" title="c2">x</font> is <a href="o_ring_1.html#V13" title="O_RING_1:attr.13">generated_from_squares</a> )</span><br/>



<b>assume </b><a NAME="E1:69"/>
<font color="Maroon" title="c2">x</font> is <a href="o_ring_1.html#V7" title="O_RING_1:attr.7">being_a_sum_of_products_of_squares</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> is <a href="o_ring_1.html#V13" title="O_RING_1:attr.13">generated_from_squares</a></span><br/>

<b>then consider </b><font color="Maroon" title="c3">f</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:69"/><i><font color="Green" title="E68">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="o_ring_1.html#V6" title="O_RING_1:attr.6">being_a_Sum_of_products_of_squares</a> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="o_ring_1.html#D9" target="_self" title="O_RING_1:def.9">Def9</a></i>;<br/>
<a NAME="E4:69"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</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> )
 
<b>by </b><i><a class="txt" href="#E3:69"><i><font color="Green" title="E68">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="E5:69"/>
<font color="Maroon" title="c2">x</font> is <a href="o_ring_1.html#V13" title="O_RING_1:attr.13">generated_from_squares</a>
 <b>by </b><i><a class="txt" href="#E3:69"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="o_ring_1.html#E69"><i><font color="Green" title="E66">Lm53</font></i></a></i>; <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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