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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="rfunct_3.html#K4" target="_self">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<a NAME="E1:47"/><i><font color="Green">E33</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D8" target="_self">Def8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><b>case </b></a><a NAME="E1:47_1_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:47_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:47_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rfunct_3.html#E1:47_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2"><b>case </b></a><a NAME="E1:47_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:47_1_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <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#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:47"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E3:47_1_1_2"/><i><font color="Green">E36</font></i>: 
(  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.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> &amp;  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp;  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</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">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><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="47_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:47_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:47_1_1_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rfunct_3.html#E2:47_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/><b>hence </b><a NAME="E3:47_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:47_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:47_1_1_2_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E3:47_1_1_2_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:47_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:47_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:47_1_1_2_1_1_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:47_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rfunct_3.html#E2:47_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="rfunct_3.html#E3:47_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rfunct_3.html#E4:47_1_1_2_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="rfunct_3.html#M2" target="_self">Element</a> of  <a href="rfunct_3.html#K4" target="_self">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="rfunct_3.html#E1:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rfunct_3.html#E3:47_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E4:47_1_1_2_1_1_2_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:47_1_1_2_1_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rfunct_3.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E5:47_1_1_2_1_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:47_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="rfunct_3.html#E4:47_1_1_2_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="rfunct_3.html#E1:47_1_1_2_1_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/><b>hence </b><a NAME="E6:47_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rfunct_3.html#E3:47_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rfunct_3.html#E1:47_1_1_2_1_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="rfunct_3.html#E4:47_1_1_2_1_1_2_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rfunct_3.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:47_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rfunct_3.html#E2:47_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:47_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:47"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K18" target="_self">#</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


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