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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#NR1" title="O_RING_1:NR.1">is_a_square</a>  &amp; <font color="Olive" title="b2">y</font> <a href="o_ring_1.html#NR1" title="O_RING_1:NR.1">is_a_square</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#NR5" title="O_RING_1:NR.5">is_a_product_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#NR1" title="O_RING_1:NR.1">is_a_square</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="o_ring_1.html#NR1" title="O_RING_1:NR.1">is_a_square</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#NR5" title="O_RING_1:NR.5">is_a_product_of_squares</a>  )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:140"/><i><font color="Green" title="E99">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="o_ring_1.html#NR1" title="O_RING_1:NR.1">is_a_square</a> 
 <b>and </b><br/><a NAME="E2:140"/><i><font color="Green" title="E100">A2</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="o_ring_1.html#NR1" title="O_RING_1:NR.1">is_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> <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#NR5" title="O_RING_1:NR.5">is_a_product_of_squares</a> </span><br/>

<a NAME="E3:140"/>
<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>by </b><i><a class="txt" href="o_ring_1.html#E1:140"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="o_ring_1.html#E16"><i><font color="Green" title="E13">Lm9</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">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="E5:140"/><i><font color="Green" title="E101">A3</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="o_ring_1.html#NR4" title="O_RING_1:NR.4">is_a_Product_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="c4">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="c4">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="o_ring_1.html#D7" target="_self" title="O_RING_1:def.7">Def7</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><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></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="o_ring_1.html#D7" target="_self" title="O_RING_1:def.7">O_RING_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">g</font> <a href="o_ring_1.html#NR4" title="O_RING_1:NR.4">is_a_Product_of_squares</a>  &amp; <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</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="c5">g</font></span>)</span> )</span><br/>

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><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></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<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="E7:140"/>
( <font color="Maroon" title="c5">g</font> <a href="o_ring_1.html#NR4" title="O_RING_1:NR.4">is_a_Product_of_squares</a>  &amp; <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</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="c5">g</font></span>)</span> )
 <b>by </b><i><a class="txt" href="o_ring_1.html#E2:140"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="o_ring_1.html#E5:140"><i><font color="Green" title="E101">A3</font></i></a>, <a class="txt" href="o_ring_1.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="txt" href="o_ring_1.html#E136"><i><font color="Green" title="E96">Lm83</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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