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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="E1:47"/>
<font color="Maroon">x</font> <a href="o_ring_1.html#NR3" target="_self">is_a_sum_of_squares</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:47"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f</font> <a href="o_ring_1.html#NR2" target="_self">is_a_Sum_of_squares</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:47"/>
( <font color="Maroon">f</font> <a href="o_ring_1.html#NR10" target="_self">is_a_Sum_of_amalgams_of_squares</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:47"/>
<font color="Maroon">x</font> <a href="o_ring_1.html#NR11" target="_self">is_a_sum_of_amalgams_of_squares</a> 
 <b>by </b><i/>;<br/>


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