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

<span class="kw">let </span><font color="Maroon" title="c1">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p2</font>, <font color="Olive" title="b2">p1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b3">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Olive" title="b2">p1</font></span>)</span></span>)</span> holds <br/> <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">p1</font></span>)</span>,<font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">p2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">p1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span> holds <br/> <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">p1</font></span>)</span>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">p2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span> implies  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">p1</font></span>)</span>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">p2</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:118"/><span class="lab"><font color="Green" title="E95">A1</font></span>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">p1</font></span>)</span>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">p2</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_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="E1:118_1"/>
( <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> or  not <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_1"><span class="kw">suppose </span></a><a NAME="E1:118_1_1"/><span class="lab"><font color="Green" title="E96">A2</font></span>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">p1</font></span>)</span>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">p2</font></span><br/><div class="add"><a NAME="E2:118_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A3</font></span>: 
 <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#D2" target="_self" title="FINSEQ_6:def.2">Def2</a></span>;<br/><a NAME="E3:118_1_1"/><span class="lab"><font color="Green" title="E98">A4</font></span>: 
<font color="Maroon" title="c3">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</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_6.html#T84" target="_self" title="FINSEQ_6:th.84">Th84</a></span>;<br/><a NAME="E4:118_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T36" title="FINSEQ_5:th.36">FINSEQ_5:36</a></span>;<br/><a NAME="E5:118_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A5</font></span>: 
not <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E6:118_1_1"/><span class="lab"><font color="Green" title="E100">A6</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">p1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T107" target="_self" title="FINSEQ_6:th.107">Th107</a></span>;<br/><a NAME="E7:118_1_1"/><span class="lab"><font color="Green" title="E101">A7</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_1_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="ref" href="finseq_5.html#T50" title="FINSEQ_5:th.50">FINSEQ_5:50</a></span>;<br/><a NAME="E8:118_1_1"/><span class="lab"><font color="Green" title="E102">A8</font></span>: 
not <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E9:118_1_1"/>
<font color="Maroon" title="c3">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font>
 ;<br/><a NAME="E10:118_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A9</font></span>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_1_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="txt" href="#E8:118_1_1"><span class="lab"><font color="Green" title="E102">A8</font></span></a>, <a class="ref" href="finseq_5.html#T49" title="FINSEQ_5:th.49">FINSEQ_5:49</a></span>;<br/><a NAME="E11:118_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_1_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="ref" href="finseq_6.html#T75" target="_self" title="FINSEQ_6:th.75">Th75</a></span>;<br/><a NAME="E12:118_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A10</font></span>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="txt" href="#E8:118_1_1"><span class="lab"><font color="Green" title="E102">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E13:118_1_1"/>
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T66" target="_self" title="FINSEQ_6:th.66">Th66</a></span>;<br/><a NAME="E14:118_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A11</font></span>: 
<font color="Maroon" title="c2">p2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118_1_1"><span class="lab"><font color="Green" title="E104">A10</font></span></a>, <a class="txt" href="#E10:118_1_1"><span class="lab"><font color="Green" title="E103">A9</font></span></a>, <a class="txt" href="#E3:118_1_1"><span class="lab"><font color="Green" title="E98">A4</font></span></a>, <a class="ref" href="finseq_5.html#T10" title="FINSEQ_5:th.10">FINSEQ_5:10</a></span>;<br/><a NAME="E15:118_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118_1_1"><span class="lab"><font color="Green" title="E104">A10</font></span></a>, <a class="ref" href="finseq_6.html#T83" target="_self" title="FINSEQ_6:th.83">Th83</a></span>;<br/><a NAME="E16:118_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A12</font></span>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:118_1_1"><span class="lab"><font color="Green" title="E99">A5</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><div><span class="lab"><font color="Green" title="E107">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:118_1_1_1"/>
<font color="Maroon" title="c2">p2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:118_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_1_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></span>;<br/><span class="kw">hence </span><a NAME="E3:118_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="txt" href="#E1:118_1_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="txt" href="#E8:118_1_1"><span class="lab"><font color="Green" title="E102">A8</font></span></a>, <a class="ref" href="finseq_5.html#T49" title="FINSEQ_5:th.49">FINSEQ_5:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E18:118_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="ref" href="finseq_6.html#T83" target="_self" title="FINSEQ_6:th.83">Th83</a></span>;<br/><span class="kw">hence </span> <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">p1</font></span>)</span>,<font color="Maroon" title="c2">p2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118_1_1"><span class="lab"><font color="Green" title="E97">A3</font></span></a>, <a class="txt" href="#E6:118_1_1"><span class="lab"><font color="Green" title="E100">A6</font></span></a>, <a class="ref" href="finseq_6.html#D2" target="_self" title="FINSEQ_6:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118_1_1"><span class="lab"><font color="Green" title="E104">A10</font></span></a>, <a class="ref" href="finseq_6.html#T69" target="_self" title="FINSEQ_6:th.69">Th69</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="txt" href="#E1:118_1_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="ref" href="finseq_6.html#T76" target="_self" title="FINSEQ_6:th.76">Th76</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118_1_1"><span class="lab"><font color="Green" title="E104">A10</font></span></a>, <a class="ref" href="finseq_6.html#T71" target="_self" title="FINSEQ_6:th.71">Th71</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118_1_1"><span class="lab"><font color="Green" title="E104">A10</font></span></a>, <a class="txt" href="#E14:118_1_1"><span class="lab"><font color="Green" title="E105">A11</font></span></a>, <a class="ref" href="finseq_6.html#T65" target="_self" title="FINSEQ_6:th.65">Th65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:118_1_1"><span class="lab"><font color="Green" title="E106">A12</font></span></a>, <a class="ref" href="finseq_6.html#T72" target="_self" title="FINSEQ_6:th.72">Th72</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:118_1_1"><span class="lab"><font color="Green" title="E101">A7</font></span></a>, <a class="ref" href="finseq_6.html#T82" target="_self" title="FINSEQ_6:th.82">Th82</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118_1_1"><span class="lab"><font color="Green" title="E96">A2</font></span></a>, <a class="ref" href="finseq_6.html#T81" target="_self" title="FINSEQ_6:th.81">Th81</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="txt" href="#E17:118_1_1"><span class="lab"><font color="Green" title="E107">A13</font></span></a>, <a class="ref" href="finseq_6.html#T65" target="_self" title="FINSEQ_6:th.65">Th65</a></span>
<br/>.= 
 <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">p2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:118"><span class="lab"><font color="Green" title="E95">A1</font></span></a>, <a class="ref" href="finseq_6.html#D2" target="_self" title="FINSEQ_6:def.2">Def2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="118_1_2"><span class="kw">suppose </span></a><a NAME="E1:118_1_2"/>
not <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">p1</font></span>)</span>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">p2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:118_1_2"/>
 <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">p1</font></span>)</span>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#D2" target="_self" title="FINSEQ_6:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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