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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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green">E32</font></i>: 
<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>and </b><br/><a NAME="E2:56"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E3:56"/><i><font color="Green">E35</font></i>: 
<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_4.html#K7" target="_self">|--</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:56"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K7" target="_self">|--</a> <font color="Maroon">x</font></span>)</span>
 <b>and </b><br/><a NAME="E6:56"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K7" target="_self">|--</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="finseq_4.html#E9:56"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_4.html#E11:56"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E8:56"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K7" target="_self">|--</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_4.html#E6:56"><i><font color="Green">E37</font></i></a>, <a class="txt" href="finseq_4.html#E11:56"><i><font color="Green">E40</font></i></a>, <a class="txt" href="finseq_4.html#E1:56"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E9:56"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="txt" href="finseq_4.html#E6:56"><i><font color="Green">E37</font></i></a>, , </i>;<br/>
<a NAME="E10:56"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="finseq_4.html#E6:56"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E11:56"/><b>then </b><i><font color="Green">E40</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:56"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">y</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> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K7" target="_self">|--</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_4.html#E11:56"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:56"/><b>then </b>
( <font color="Maroon">y</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">p</font> <a href="finseq_4.html#K7" target="_self">|--</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K7" target="_self">|--</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="finseq_4.html#E6:56"><i><font color="Green">E37</font></i></a>, <a class="txt" href="finseq_4.html#E1:56"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:56"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E15:56"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span> <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">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E16:56"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:56"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_4.html#E8:56"><i><font color="Green">E38</font></i></a>, <a class="txt" href="finseq_4.html#E12:56"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_4.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_4.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E18:56"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
;<br/>
<b>hence </b><a NAME="E19:56"/>
contradiction
 <b>by </b><i><a class="ref" href="finseq_4.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


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