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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:49"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:49"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K13" target="_self">IDEAoperationC</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:49"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K13" target="_self">IDEAoperationC</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<a NAME="E6:49"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E43</font></i>:  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</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="idea_1.html#E3:49"><i><font color="Green">E40</font></i></a>, <a class="ref" href="idea_1.html#D13" target="_self">Def13</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</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="idea_1.html#E5:49"><i><font color="Green">E41</font></i></a>, <a class="ref" href="idea_1.html#D13" target="_self">Def13</a></i>
<br/>.= 
 <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_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E8:49"/>
3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:49"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:49"/><b>then </b>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:49"/><b>then </b>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:49"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> 2
 
<b>by </b><i><a class="txt" href="idea_1.html#E3:49"><i><font color="Green">E40</font></i></a>, <a class="txt" href="idea_1.html#E45"><i><font color="Green">Lemma34</font></i></a></i>;<br/>
<a NAME="E12:49"/>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:49"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:49"/><b>then </b>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:49"/><b>then </b>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:49"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> 3
 
<b>by </b><i><a class="txt" href="idea_1.html#E3:49"><i><font color="Green">E40</font></i></a>, <a class="txt" href="idea_1.html#E44"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E2:49_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</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="idea_1.html#E3:49"><i><font color="Green">E40</font></i></a>, <a class="ref" href="idea_1.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E3:49_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</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_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:49_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:49_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E5:49"><i><font color="Green">E41</font></i></a>, <a class="txt" href="idea_1.html#E11:49"><i><font color="Green">E44</font></i></a>, <a class="txt" href="idea_1.html#E3:49_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="idea_1.html#E44"><i><font color="Green">Lemma33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E5:49"><i><font color="Green">E41</font></i></a>, <a class="txt" href="idea_1.html#E15:49"><i><font color="Green">E45</font></i></a>, <a class="txt" href="idea_1.html#E3:49_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="idea_1.html#E45"><i><font color="Green">Lemma34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_3"><b>suppose </b></a><a NAME="E1:49_2_1_1_3"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 )
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="idea_1.html#E5:49"><i><font color="Green">E41</font></i></a>, <a class="txt" href="idea_1.html#E3:49_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="idea_1.html#E46"><i><font color="Green">Lemma35</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="idea_1.html#E3:49"><i><font color="Green">E40</font></i></a>, <a class="txt" href="idea_1.html#E4:49_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="idea_1.html#E1:49_2_1_1_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="idea_1.html#E46"><i><font color="Green">Lemma35</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:49_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:49"/>
 <a href="idea_1.html#K13" target="_self">IDEAoperationC</a> <span class="p1">(<span class="default"><a href="idea_1.html#K13" target="_self">IDEAoperationC</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E3:49"><i><font color="Green">E40</font></i></a>, <a class="txt" href="idea_1.html#E5:49"><i><font color="Green">E41</font></i></a>, <a class="txt" href="idea_1.html#E6:49"><i><font color="Green">E42</font></i></a>, <a class="txt" href="idea_1.html#E7:49"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


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