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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#V1" title="O_RING_1:attr.1">being_a_square</a> holds <br/><font color="Olive" title="b1">x</font> is <a href="o_ring_1.html#V11" title="O_RING_1:attr.11">being_a_sum_of_amalgams_of_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#V1" title="O_RING_1:attr.1">being_a_square</a> implies <font color="Maroon" title="c2">x</font> is <a href="o_ring_1.html#V11" title="O_RING_1:attr.11">being_a_sum_of_amalgams_of_squares</a> )</span><br/>



<b>assume </b><a NAME="E1:33"/>
<font color="Maroon" title="c2">x</font> is <a href="o_ring_1.html#V1" title="O_RING_1:attr.1">being_a_square</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#V11" title="O_RING_1:attr.11">being_a_sum_of_amalgams_of_squares</a></span><br/>

<a NAME="E2:33"/><b>then </b><i><font color="Green" title="E32">A1</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> is <a href="o_ring_1.html#V10" title="O_RING_1:attr.10">being_a_Sum_of_amalgams_of_squares</a>
 
<b>by </b><i><a class="txt" href="o_ring_1.html#E33"><i><font color="Green" title="E30">Lm20</font></i></a></i>;<br/>
<a NAME="E3:33"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<b>by </b><i><a class="txt" href="o_ring_1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:33"/>
<font color="Maroon" title="c2">x</font> is <a href="o_ring_1.html#V11" title="O_RING_1:attr.11">being_a_sum_of_amalgams_of_squares</a>
 <b>by </b><i><a class="txt" href="#E2:33"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="o_ring_1.html#D13" target="_self" title="O_RING_1:def.13">Def13</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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