<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/> for <font color="Olive" title="b3">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  not <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">x</font> &amp;  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b2">y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> implies  ex <font color="Olive" title="b1">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/> for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> &amp;  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">y</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">and </span><br/><a NAME="E2:53"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/> for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> &amp;  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">y</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">x</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:53"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font>
 <span class="kw">and </span><br/><a NAME="E6:53"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E7:53"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="jordan2c.html#T46" target="_self" title="JORDAN2C:th.46">Th54</a></span>;<br/>
<a NAME="E8:53"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E9:53"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A7</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:53_1"/>
( <font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><span class="kw">suppose </span></a><a NAME="E1:53_1_1"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
<font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/> for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> &amp;  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">y</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">g</font> = <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>)</span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><span class="lab"><font color="Green" title="E51">A9</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E9:53"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
;<br/><a NAME="E4:53_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A10</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> = 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53_1_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>

;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">y2</font> = <font color="Maroon" title="c5">g</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="jordan2c.html#T44" target="_self" title="JORDAN2C:th.44">Th52</a></span>;<br/><a NAME="E7:53_1_1"/><span class="lab"><font color="Green" title="E53">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c7">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><a NAME="E2:53_1_1_3"/><span class="lab"><font color="Green" title="E54">A12</font></span>: 
( <font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:53_1_1_3_1"/>
( <font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1_1_3"><span class="lab"><font color="Green" title="E54">A12</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:53_1_1_3_1_1"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
<font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:53_1_1_3_1_1"/>
<font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E3:53_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A14</font></span>: 
<font color="Maroon" title="c4">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E4:53_1_1_3_1_1"/><span class="lab"><font color="Green" title="E57">A15</font></span>: 
( <font color="Maroon" title="c4">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E5:53_1_1_3_1_1"/><span class="lab"><font color="Green" title="E58">A16</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:53_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A17</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53_1_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E7:53_1_1_3_1_1"/><span class="lab"><font color="Green" title="E60">A18</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E5:53_1_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><span class="lab"><font color="Green" title="E61">A19</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<font color="Maroon" title="c4">i2</font>

;<br/><a NAME="E9:53_1_1_3_1_1"/><span class="lab"><font color="Green" title="E62">A20</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E10:53_1_1_3_1_1"/>
( 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>)</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E1:53_1_1"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E3:53_1_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font> = 
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53_1_1"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E3:53_1_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E5:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E4:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="txt" href="#E3:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E8:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E61">A19</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
;<br/><a NAME="E12:53_1_1_3_1_1"/><span class="kw">then </span>
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E1:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E7:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E9:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E62">A20</font></span></a>, <a class="ref" href="jordan2c.html#T45" target="_self" title="JORDAN2C:th.45">Th53</a></span>;<br/><a NAME="E13:53_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A21</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E9:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E62">A20</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><a NAME="E14:53_1_1_3_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E5:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="jordan2c.html#T45" target="_self" title="JORDAN2C:th.45">Th53</a></span>;<br/><a NAME="E15:53_1_1_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="txt" href="#E6:53_1_1_3_1_1"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>;<br/><span class="kw">hence </span><a NAME="E16:53_1_1_3_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:53_1_1_3_1_2"/><span class="lab"><font color="Green" title="E55">A22</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:53_1_1_3_1_2"/>
<font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E3:53_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A23</font></span>: 
<font color="Maroon" title="c4">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E4:53_1_1_3_1_2"/><span class="lab"><font color="Green" title="E57">A24</font></span>: 
( <font color="Maroon" title="c4">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_1"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E5:53_1_1_3_1_2"/><span class="lab"><font color="Green" title="E58">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:53_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A26</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53_1_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E7:53_1_1_3_1_2"/><span class="lab"><font color="Green" title="E60">A27</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E5:53_1_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><span class="lab"><font color="Green" title="E61">A28</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<font color="Maroon" title="c4">i2</font>

;<br/><a NAME="E9:53_1_1_3_1_2"/><span class="lab"><font color="Green" title="E62">A29</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E10:53_1_1_3_1_2"/>
( 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>)</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E1:53_1_1"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E3:53_1_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font> = 
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53_1_1"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E3:53_1_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E5:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="txt" href="#E4:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E57">A24</font></span></a>, <a class="txt" href="#E3:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E56">A23</font></span></a>, <a class="txt" href="#E8:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E61">A28</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
;<br/><a NAME="E12:53_1_1_3_1_2"/><span class="kw">then </span>
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E5:53_1_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="txt" href="#E1:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E55">A22</font></span></a>, <a class="txt" href="#E7:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E60">A27</font></span></a>, <a class="txt" href="#E9:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="ref" href="jordan2c.html#T45" target="_self" title="JORDAN2C:th.45">Th53</a></span>;<br/><a NAME="E13:53_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A30</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E9:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><a NAME="E14:53_1_1_3_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53_1_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="txt" href="#E1:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E55">A22</font></span></a>, <a class="txt" href="#E5:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="ref" href="jordan2c.html#T45" target="_self" title="JORDAN2C:th.45">Th53</a></span>;<br/><a NAME="E15:53_1_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E63">A30</font></span></a>, <a class="txt" href="#E6:53_1_1_3_1_2"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>;<br/><span class="kw">hence </span><a NAME="E16:53_1_1_3_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:53_1_1"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/> for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> &amp;  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><span class="kw">suppose </span></a><a NAME="E1:53_1_2"/><span class="lab"><font color="Green" title="E50">A31</font></span>: 
<font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/> for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> &amp;  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">y</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">g</font> = <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><a NAME="E3:53_1_2"/><span class="lab"><font color="Green" title="E51">A32</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><a NAME="E4:53_1_2"/><span class="lab"><font color="Green" title="E52">A33</font></span>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E5:53_1_2"/><span class="lab"><font color="Green" title="E53">A34</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A35</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_1_2"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1

;<br/><a NAME="E7:53_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A36</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:53"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E5:53_1_2"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E3:53_1_2"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E4:53_1_2"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><a NAME="E8:53_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:53"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E5:53_1_2"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E3:53_1_2"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E4:53_1_2"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E6:53_1_2"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A37</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>

;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">y2</font> = <font color="Maroon" title="c5">g</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="jordan2c.html#T44" target="_self" title="JORDAN2C:th.44">Th52</a></span>;<br/><a NAME="E11:53_1_2"/><span class="lab"><font color="Green" title="E57">A38</font></span>: 
<font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E1:53_1_2"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c7">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><a NAME="E2:53_1_2_3"/><span class="lab"><font color="Green" title="E58">A39</font></span>: 
( <font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:53_1_2_3_1"/>
( <font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1_2_3"><span class="lab"><font color="Green" title="E58">A39</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:53_1_2_3_1_1"/><span class="lab"><font color="Green" title="E59">A40</font></span>: 
<font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:53_1_2_3_1_1"/><span class="lab"><font color="Green" title="E60">A41</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E9:53_1_2"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E3:53_1_2_3_1_1"/><span class="lab"><font color="Green" title="E61">A42</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font> = 
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53_1_2"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E4:53_1_2"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E7:53_1_2"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="ref" href="finseq_6.html#T109" title="FINSEQ_6:th.109">FINSEQ_6:109</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53_1_2"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E5:53_1_2"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E3:53_1_2"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E4:53_1_2"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="ref" href="finseq_6.html#T123" title="FINSEQ_6:th.123">FINSEQ_6:123</a></span>
;<br/><a NAME="E5:53_1_2_3_1_1"/><span class="kw">then </span>
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E1:53_1_2_3_1_1"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="txt" href="#E2:53_1_2_3_1_1"><span class="lab"><font color="Green" title="E60">A41</font></span></a>, <a class="txt" href="#E3:53_1_2_3_1_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="ref" href="jordan2c.html#T45" target="_self" title="JORDAN2C:th.45">Th53</a></span>;<br/><a NAME="E6:53_1_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A43</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E3:53_1_2_3_1_1"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><span class="lab"><font color="Green" title="E63">A44</font></span>: <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
<font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53_1_2"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></span>
;<br/><a NAME="E8:53_1_2_3_1_1"/><span class="lab"><font color="Green" title="E64">A45</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:53_1_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A46</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:53_1_2"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E10:53_1_2_3_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1_2_3_1_1"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="txt" href="#E8:53_1_2_3_1_1"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="ref" href="jordan2c.html#T45" target="_self" title="JORDAN2C:th.45">Th53</a></span>;<br/><span class="kw">hence </span><a NAME="E11:53_1_2_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53_1_2_3_1_1"><span class="lab"><font color="Green" title="E62">A43</font></span></a>, <a class="txt" href="#E9:53_1_2_3_1_1"><span class="lab"><font color="Green" title="E65">A46</font></span></a>, <a class="txt" href="#E7:53_1_2_3_1_1"><span class="lab"><font color="Green" title="E63">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:53_1_2_3_1_2"/><span class="lab"><font color="Green" title="E59">A47</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:53_1_2_3_1_2"/><span class="lab"><font color="Green" title="E60">A48</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E9:53_1_2"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E3:53_1_2_3_1_2"/><span class="lab"><font color="Green" title="E61">A49</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font> = 
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53_1_2"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E4:53_1_2"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E7:53_1_2"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="ref" href="finseq_6.html#T109" title="FINSEQ_6:th.109">FINSEQ_6:109</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53_1_2"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E5:53_1_2"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E3:53_1_2"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E4:53_1_2"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="ref" href="finseq_6.html#T123" title="FINSEQ_6:th.123">FINSEQ_6:123</a></span>
;<br/><a NAME="E5:53_1_2_3_1_2"/><span class="kw">then </span>
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E9:53_1_2"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="txt" href="#E1:53_1_2_3_1_2"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="txt" href="#E2:53_1_2_3_1_2"><span class="lab"><font color="Green" title="E60">A48</font></span></a>, <a class="txt" href="#E3:53_1_2_3_1_2"><span class="lab"><font color="Green" title="E61">A49</font></span></a>, <a class="ref" href="jordan2c.html#T45" target="_self" title="JORDAN2C:th.45">Th53</a></span>;<br/><a NAME="E6:53_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A50</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E3:53_1_2_3_1_2"><span class="lab"><font color="Green" title="E61">A49</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><span class="lab"><font color="Green" title="E63">A51</font></span>: <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
<font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53_1_2"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></span>
;<br/><a NAME="E8:53_1_2_3_1_2"/><span class="lab"><font color="Green" title="E64">A52</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:53_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A53</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:53_1_2"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><a NAME="E10:53_1_2_3_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:53_1_2"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="txt" href="#E1:53_1_2_3_1_2"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="txt" href="#E8:53_1_2_3_1_2"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="ref" href="jordan2c.html#T45" target="_self" title="JORDAN2C:th.45">Th53</a></span>;<br/><span class="kw">hence </span><a NAME="E11:53_1_2_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53_1_2_3_1_2"><span class="lab"><font color="Green" title="E62">A50</font></span></a>, <a class="txt" href="#E9:53_1_2_3_1_2"><span class="lab"><font color="Green" title="E65">A53</font></span></a>, <a class="txt" href="#E7:53_1_2_3_1_2"><span class="lab"><font color="Green" title="E63">A51</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:53_1_2"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/> for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> (  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> &amp;  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
