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<b>let </b><font color="Maroon" title="c1">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="c2">x</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Scalar</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>assume </b><a NAME="E1:31"/>
<font color="Maroon" title="c2">x</font> <a href="o_ring_1.html#NR3" title="O_RING_1:NR.3">is_a_sum_of_squares</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">f</font> being    <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="c1">R</font><b> such that </b><br/><a NAME="E3:31"/><i><font color="Green" title="E23">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>  &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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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#D5" target="_self" title="O_RING_1:def.5">Def5</a></i>;<br/>
<a NAME="E4:31"/>
<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>by </b><i><a class="txt" href="o_ring_1.html#E3:31"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="o_ring_1.html#E24"><i><font color="Green" title="E21">Lm15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:31"/>
<font color="Maroon" title="c2">x</font> <a href="o_ring_1.html#NR7" title="O_RING_1:NR.7">is_a_sum_of_products_of_squares</a> 
 <b>by </b><i><a class="txt" href="o_ring_1.html#E3:31"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="o_ring_1.html#D9" target="_self" title="O_RING_1:def.9">Def9</a></i>;<br/>


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