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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">f</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">F<sub>3</sub></font>() st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F<sub>4</sub></font>() &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) );<br/>
<a NAME="E1:8_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F<sub>4</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i/>;<br/>
<a NAME="E2:8_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:8_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>consider </b><font color="Maroon">g</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E6:8_1"/><i><font color="Green">E36</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F<sub>4</sub></font>() &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1"/>
not <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:8_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</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/>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:8_1_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>set </b><font color="Maroon">d</font> =  <a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font>;<br/><a NAME="E4:8_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:8_1_1_1_1"/><i><font color="Green">E52</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:8_1_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:8_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:8_1_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:8_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:8_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E11:8_1_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E12:8_1_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_1_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:8_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E4:8_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/><a NAME="E5:8_1_1_1_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E6:8_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:8_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E3:8_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E4:8_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/></div><b>end;</b></div><a NAME="E8:8_1_1_1_1_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span>
 ;<br/><a NAME="E9:8_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:8_1_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E11:8_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E12:8_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E13:8_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E14:8_1_1_1_1_1_1_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span>
 <b>by </b><i>, , , , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_1_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_1_1_1_2"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E3:8_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>()
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E13:8_1_1_1_1"/><i><font color="Green">E75</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F<sub>4</sub></font>() &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:8_1_1_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:8_1_1_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:8_1_1_1_1_2"/><i><font color="Green">E76</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> )
 ;<br/>

<a NAME="E4:8_1_1_1_1_2"/><i><font color="Green">E77</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1_1_2_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_2_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E3:8_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:8_1_1_1_1_2_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:8_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:8_1_1_1_1_2_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:8_1_1_1_1_2_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_2_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E3:8_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_1_2_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:8_1_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1_2_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E5:8_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:8_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E7:8_1_1_1_1_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E8:8_1_1_1_1_2_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <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">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:8_1_1_1_1_2_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:8_1_1_1_1_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E11:8_1_1_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E12:8_1_1_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><b>then </b><span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">g</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> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
;<br/><b>hence </b><a NAME="E14:8_1_1_1_1_2_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_2_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_1_2_1_3"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:8_1_1_1_1_2_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , , </i>;<br/><a NAME="E4:8_1_1_1_1_2_1_3"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E5:8_1_1_1_1_2_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:8_1_1_1_1_2_1_3"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:8_1_1_1_1_2_1_3"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E8:8_1_1_1_1_2_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , , </i>;<br/><a NAME="E9:8_1_1_1_1_2_1_3"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E10:8_1_1_1_1_2_1_3"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E11:8_1_1_1_1_2_1_3"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:8_1_1_1_1_2_1_3"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E13:8_1_1_1_1_2_1_3"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><b>then </b><i><font color="Green">E91</font></i>: <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E15:8_1_1_1_1_2_1_3"/><i><font color="Green">E92</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E16:8_1_1_1_1_2_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E17:8_1_1_1_1_2_1_3"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E18:8_1_1_1_1_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E19:8_1_1_1_1_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E20:8_1_1_1_1_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>hence </b><a NAME="E21:8_1_1_1_1_2_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i>, , <a class="ref" href="pencil_2.html#T1" target="_self">Th1</a>, <a class="txt" href="pencil_2.html#E1:8"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E14:8_1_1_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E15:8_1_1_1_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E16:8_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E17:8_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E18:8_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E19:8_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:8_1_1_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>set </b><font color="Maroon">d</font> =  <a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font>;<br/><a NAME="E4:8_1_1_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:8_1_1_1_2"/><i><font color="Green">E97</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:8_1_1_1_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:8_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:8_1_1_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:8_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:8_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E11:8_1_1_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E12:8_1_1_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_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="E1:8_1_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_2_1_1_1"/><i><font color="Green">E102</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:8_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E4:8_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/><a NAME="E5:8_1_1_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E6:8_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_2_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:8_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E3:8_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E4:8_1_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/></div><b>end;</b></div><a NAME="E8:8_1_1_1_2_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span>
 ;<br/><a NAME="E9:8_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E104</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:8_1_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E11:8_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E12:8_1_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E13:8_1_1_1_2_1_1_1"/><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"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E14:8_1_1_1_2_1_1_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i>, , , , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_2_1_1_2"/><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_2_1_1_2"/><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E3:8_1_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>()
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E13:8_1_1_1_2"/><i><font color="Green">E108</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F<sub>4</sub></font>() &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:8_1_1_1_2_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:8_1_1_1_2_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:8_1_1_1_2_2"/><i><font color="Green">E109</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> )
 ;<br/>

<a NAME="E4:8_1_1_1_2_2"/><i><font color="Green">E110</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1_2_2_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">y</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_2_2_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E3:8_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:8_1_1_1_2_2_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:8_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:8_1_1_1_2_2_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:8_1_1_1_2_2_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_2_2_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E3:8_1_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1_1_2_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:8_1_1_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_2_2_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E5:8_1_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:8_1_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E7:8_1_1_1_2_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E8:8_1_1_1_2_2_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <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">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:8_1_1_1_2_2_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:8_1_1_1_2_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E11:8_1_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E12:8_1_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><b>then </b><span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
;<br/><b>hence </b><a NAME="E14:8_1_1_1_2_2_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:8_1_1_1_2_2_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_2_2_1_3"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:8_1_1_1_2_2_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , , </i>;<br/><a NAME="E4:8_1_1_1_2_2_1_3"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E5:8_1_1_1_2_2_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:8_1_1_1_2_2_1_3"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:8_1_1_1_2_2_1_3"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E8:8_1_1_1_2_2_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , , </i>;<br/><a NAME="E9:8_1_1_1_2_2_1_3"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E10:8_1_1_1_2_2_1_3"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E11:8_1_1_1_2_2_1_3"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:8_1_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E13:8_1_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><b>then </b><i><font color="Green">E124</font></i>: <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E15:8_1_1_1_2_2_1_3"/><i><font color="Green">E125</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E16:8_1_1_1_2_2_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E17:8_1_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E18:8_1_1_1_2_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E19:8_1_1_1_2_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E20:8_1_1_1_2_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>hence </b><a NAME="E21:8_1_1_1_2_2_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i>, , <a class="ref" href="pencil_2.html#T1" target="_self">Th1</a>, <a class="txt" href="pencil_2.html#E1:8"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E14:8_1_1_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E15:8_1_1_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E16:8_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E17:8_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E18:8_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pencil_2.html#K1" target="_self">Del</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E19:8_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:8_1"/>
 ex <font color="Olive">f</font> being <a href="funct_1.html#V2">one-to-one</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">F<sub>3</sub></font>() st <br/>( <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F<sub>4</sub></font>() &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>by </b><i/>;<br/>


</div>
