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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> is <a href="o_ring_1.html#V7" title="O_RING_1:attr.7">being_a_sum_of_products_of_squares</a> &amp; <font color="Olive" title="b2">y</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> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> is <a href="o_ring_1.html#V7" title="O_RING_1:attr.7">being_a_sum_of_products_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> is <a href="o_ring_1.html#V7" title="O_RING_1:attr.7">being_a_sum_of_products_of_squares</a> &amp; <font color="Maroon" title="c3">y</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> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">y</font> is <a href="o_ring_1.html#V7" title="O_RING_1:attr.7">being_a_sum_of_products_of_squares</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:80"/><i><font color="Green" title="E79">A1</font></i>: 
<font color="Maroon" title="c2">x</font> is <a href="o_ring_1.html#V7" title="O_RING_1:attr.7">being_a_sum_of_products_of_squares</a>
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green" title="E80">A2</font></i>: 
<font color="Maroon" title="c3">y</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> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">y</font> is <a href="o_ring_1.html#V7" title="O_RING_1:attr.7">being_a_sum_of_products_of_squares</a></span><br/>

<a NAME="E3:80"/><i><font color="Green" title="E81">A3</font></i>: 
<font color="Maroon" title="c3">y</font> is <a href="o_ring_1.html#V5" title="O_RING_1:attr.5">being_a_product_of_squares</a>
 
<b>by </b><i><a class="txt" href="#E2:80"><i><font color="Green" title="E80">A2</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>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="E5:80"/><i><font color="Green" title="E82">A4</font></i>: 
( <font color="Maroon" title="c4">f</font> is <a href="o_ring_1.html#V6" title="O_RING_1:attr.6">being_a_Sum_of_products_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="txt" href="#E1:80"><i><font color="Green" title="E79">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/>
<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#K1" title="ALGSTR_0:func.1">+</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#D9" target="_self" title="O_RING_1:def.9">O_RING_1:def 9</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> is <a href="o_ring_1.html#V6" title="O_RING_1:attr.6">being_a_Sum_of_products_of_squares</a> &amp; <font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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#K1" title="ALGSTR_0:func.1">+</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="E7:80"/>
( <font color="Maroon" title="c5">g</font> is <a href="o_ring_1.html#V6" title="O_RING_1:attr.6">being_a_Sum_of_products_of_squares</a> &amp; <font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="#E3:80"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E5:80"><i><font color="Green" title="E82">A4</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#E78"><i><font color="Green" title="E70">Lm57</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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