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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>let </b><font color="Maroon">y</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="o_ring_1.html#NR11" target="_self">is_a_sum_of_amalgams_of_squares</a> 
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">y</font> <a href="o_ring_1.html#NR13" target="_self">is_generated_from_squares</a> 
 ;<br/>

<a NAME="E3:62"/>
<font color="Maroon">x</font> <a href="o_ring_1.html#NR13" target="_self">is_generated_from_squares</a> 
 
<b>by </b><i><a class="txt" href="o_ring_3.html#E18"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="o_ring_1.html#T6">O_RING_1:6</a></i>;<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="E5:62"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> <a href="o_ring_1.html#NR12" target="_self">is_a_generation_from_squares</a> 
 <b>and </b><br/><a NAME="E6:62"/><i><font color="Green">E30</font></i>: 
<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><a class="ref" href="o_ring_1.html#D15">O_RING_1:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon">g</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="E8:62"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">g</font> <a href="o_ring_1.html#NR12" target="_self">is_a_generation_from_squares</a> 
 <b>and </b><br/><a NAME="E9:62"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</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">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="o_ring_3.html#E19"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="o_ring_1.html#D15">O_RING_1:def 15</a></i>;<br/>
<b>set </b><font color="Maroon">h</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E10:62"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="o_ring_1.html#NR12" target="_self">is_a_generation_from_squares</a> 
 
<b>by </b><i><a class="txt" href="o_ring_3.html#E20"><i><font color="Green">Lemma66</font></i></a>, <a class="txt" href="o_ring_3.html#E22"><i><font color="Green">Lemma68</font></i></a>, </i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i/>
;<br/>
<a NAME="E12:62"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="o_ring_3.html#E21"><i><font color="Green">Lemma67</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E13:62"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="o_ring_1.html#NR13" target="_self">is_generated_from_squares</a> 
 <b>by </b><i><a class="txt" href="o_ring_3.html#E23"><i><font color="Green">Lemma69</font></i></a>, <a class="txt" href="o_ring_3.html#E26"><i><font color="Green">Lemma70</font></i></a>, <a class="ref" href="o_ring_1.html#D15">O_RING_1:def 15</a></i>;<br/>


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