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<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:56_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:56_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:56_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" target="_self">Th4</a>, </i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E6:56_1"/><i><font color="Green">E34</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="finseq_1.html#E4:56_1"><i><font color="Green">E33</font></i></a>, </i>;<br/><div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:56_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" target="_self">Th4</a>, <a class="txt" href="finseq_1.html#E6:56_1"><i><font color="Green">E34</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:56_1_1"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E1:56_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:56_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E7:56_1_1"/><i><font color="Green">E53</font></i>: 
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E8:56_1_1"/>
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E6:56_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:56_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E10:56_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5" target="_self">Th5</a>, </i>;<br/><b>hence </b><a NAME="E11:56_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:56_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1_2"/>
not <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:56_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:56_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E6:56_1"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E4:56_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E5:56_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" target="_self">Th4</a>, </i>;<br/><b>hence </b><a NAME="E6:56_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:56_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_1.html#E7:56_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:56"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:56"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E5:56"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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