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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>, <font color="Olive" title="b2">y</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> <a href="o_ring_1.html#NR5" title="O_RING_1:NR.5">is_a_product_of_squares</a>  &amp; <font color="Olive" title="b2">y</font> <a href="o_ring_1.html#NR5" title="O_RING_1:NR.5">is_a_product_of_squares</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="o_ring_1.html#NR9" title="O_RING_1:NR.9">is_an_amalgam_of_squares</a> </span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</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> <a href="o_ring_1.html#NR5" title="O_RING_1:NR.5">is_a_product_of_squares</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="o_ring_1.html#NR5" title="O_RING_1:NR.5">is_a_product_of_squares</a>  implies <font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font> <a href="o_ring_1.html#NR9" title="O_RING_1:NR.9">is_an_amalgam_of_squares</a>  )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:144"/><i><font color="Green" title="E100">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="o_ring_1.html#NR5" title="O_RING_1:NR.5">is_a_product_of_squares</a> 
 <b>and </b><br/><a NAME="E2:144"/><i><font color="Green" title="E101">A2</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="o_ring_1.html#NR5" title="O_RING_1:NR.5">is_a_product_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> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font> <a href="o_ring_1.html#NR9" title="O_RING_1:NR.9">is_an_amalgam_of_squares</a> </span><br/>

<a NAME="E3:144"/><i><font color="Green" title="E102">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="o_ring_1.html#NR9" title="O_RING_1:NR.9">is_an_amalgam_of_squares</a> 
 
<b>by </b><i><a class="txt" href="o_ring_1.html#E1:144"><i><font color="Green" title="E100">A1</font></i></a>, <a class="txt" href="o_ring_1.html#E30"><i><font color="Green" title="E27">Lm19</font></i></a></i>;<br/>
<a NAME="E4:144"/>
<font color="Maroon" title="c3">y</font> <a href="o_ring_1.html#NR9" title="O_RING_1:NR.9">is_an_amalgam_of_squares</a> 
 
<b>by </b><i><a class="txt" href="o_ring_1.html#E2:144"><i><font color="Green" title="E101">A2</font></i></a>, <a class="txt" href="o_ring_1.html#E30"><i><font color="Green" title="E27">Lm19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:144"/>
<font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">y</font> <a href="o_ring_1.html#NR9" title="O_RING_1:NR.9">is_an_amalgam_of_squares</a> 
 <b>by </b><i><a class="txt" href="o_ring_1.html#E3:144"><i><font color="Green" title="E102">A3</font></i></a>, <a class="txt" href="o_ring_1.html#E141"><i><font color="Green" title="E99">Lm86</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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