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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="o_ring_1.html#NR12" target="_self">is_a_generation_from_squares</a> 
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="o_ring_1.html#NR12" target="_self">is_a_generation_from_squares</a> 
 ;<br/>

<a NAME="E3:23"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>3</sub></font> <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#E1:23"><i><font color="Green">E24</font></i></a>, <a class="txt" href="o_ring_3.html#E2:23"><i><font color="Green">E25</font></i></a>, <a class="txt" href="o_ring_3.html#E14"><i><font color="Green">Lemma14</font></i></a></i>;<br/>
<a NAME="E4:23"/><i><font color="Green">E27</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="o_ring_3.html#E1:23"><i><font color="Green">E24</font></i></a>, <a class="ref" href="o_ring_1.html#D14">O_RING_1:def 14</a></i>;<br/>
<a NAME="E5:23"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:23"/><b>then </b><i><font color="Green">E28</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="o_ring_3.html#E1:23"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="o_ring_1.html#D14">O_RING_1:def 14</a></i>;<br/>
<a NAME="E7:23"/><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="o_ring_3.html#E2:23"><i><font color="Green">E25</font></i></a>, <a class="ref" href="o_ring_1.html#D14">O_RING_1:def 14</a></i>;<br/>
<a NAME="E8:23"/><b>then </b><i><font color="Green">E30</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/>
<a NAME="E9:23"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="o_ring_3.html#E4:23"><i><font color="Green">E27</font></i></a>, <a class="txt" href="o_ring_3.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E10:23"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:23"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="o_ring_3.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<a NAME="E12:23"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="o_ring_3.html#E7:23"><i><font color="Green">E29</font></i></a>, <a class="txt" href="o_ring_3.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E13:23"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:23"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="o_ring_3.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:23"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></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#E3:23"><i><font color="Green">E26</font></i></a>, <a class="txt" href="o_ring_3.html#E6:23"><i><font color="Green">E28</font></i></a>, <a class="txt" href="o_ring_3.html#E8:23"><i><font color="Green">E30</font></i></a>, <a class="txt" href="o_ring_3.html#E11:23"><i><font color="Green">E31</font></i></a>, <a class="txt" href="o_ring_3.html#E18"><i><font color="Green">Lemma18</font></i></a></i>;<br/>


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