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<b>let </b><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> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:113_1"/>
( <font color="Maroon" title="c2">p</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="c3">f</font> or  not <font color="Maroon" title="c2">p</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="c3">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1"><b>suppose </b></a><a NAME="E1:113_1_1"/><i><font color="Green" title="E96">A1</font></i>: 
<font color="Maroon" title="c2">p</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="c3">f</font>
 ;<br/><div class="add"><a NAME="E2:113_1_1"/><b>then </b>
 <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p</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="c3">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_6.html#D2" target="_self" title="FINSEQ_6:def.2">Def2</a></i>;<br/><a NAME="E3:113_1_1"/><b>then </b><i><font color="Green" title="E97">A2</font></i>: 
 <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="c3">f</font>,<font color="Maroon" title="c2">p</font></span>)</span> <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="c3">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p</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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/><b>thus </b><a NAME="E4:113_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</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="c3">f</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:113_1_1_1"/><i><font color="Green" title="E98">A3</font></i>: 
<font color="Maroon" title="c4">x</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="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:113_1_1_1"/><i><font color="Green" title="E99">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</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> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T36" title="FINSEQ_5:th.36">FINSEQ_5:36</a></i>;<br/>
<a NAME="E3:113_1_1_1"/><b>then </b><i><font color="Green" title="E100">A5</font></i>: 
<font color="Maroon" title="c4">x</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="c3">f</font>
 
<b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1_1"><i><font color="Green" title="E98">A3</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:113_1_1_1_1"/>
( ( <font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> ) or ( <font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> ) or <font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:113_1_1_1_1_1"/><i><font color="Green" title="E101">A6</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E2:113_1_1_1_1_1"/><i><font color="Green" title="E102">A7</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:113_1_1_1_1_1"/><i><font color="Green" title="E103">A8</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1_1"><i><font color="Green" title="E98">A3</font></i></a>, <a class="txt" href="finseq_6.html#E2:113_1_1_1"><i><font color="Green" title="E99">A4</font></i></a>, <a class="txt" href="finseq_6.html#E1:113_1_1_1_1_1"><i><font color="Green" title="E101">A6</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/><a NAME="E4:113_1_1_1_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1"><i><font color="Green" title="E96">A1</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/><a NAME="E5:113_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="finseq_6.html#E3:113_1_1_1_1_1"><i><font color="Green" title="E103">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:113_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E104">A9</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1"><i><font color="Green" title="E96">A1</font></i></a>, <a class="ref" href="finseq_6.html#T61" target="_self" title="FINSEQ_6:th.61">Th61</a></i>;<br/><a NAME="E7:113_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1"><i><font color="Green" title="E96">A1</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/><a NAME="E8:113_1_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E9:113_1_1_1_1_1"/><b>then </b>
<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/><a NAME="E10:113_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E105">A10</font></i>: 
 <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> <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"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E11:113_1_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1_1"><i><font color="Green" title="E98">A3</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/><a NAME="E12:113_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E106">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="finseq_6.html#E10:113_1_1_1_1_1"><i><font color="Green" title="E105">A10</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E13:113_1_1_1_1_1"/><i><font color="Green" title="E107">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E2:113_1_1_1_1_1"><i><font color="Green" title="E102">A7</font></i></a>, <a class="txt" href="finseq_6.html#E6:113_1_1_1_1_1"><i><font color="Green" title="E104">A9</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E14:113_1_1_1_1_1"/><i><font color="Green" title="E108">A13</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1_1"><i><font color="Green" title="E98">A3</font></i></a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a></i>;<br/><a NAME="E15:113_1_1_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> <b>by </b><i><a class="txt" href="finseq_6.html#E3:113_1_1_1"><i><font color="Green" title="E100">A5</font></i></a></i>;<br/><font color="Maroon" title="c5">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1_1"><i><font color="Green" title="E98">A3</font></i></a>, <a class="ref" href="finseq_5.html#T41" title="FINSEQ_5:th.41">FINSEQ_5:41</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="finseq_6.html#E14:113_1_1_1_1_1"><i><font color="Green" title="E108">A13</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/><a NAME="E18:113_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</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="c3">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1"><i><font color="Green" title="E96">A1</font></i></a>, <a class="txt" href="finseq_6.html#E12:113_1_1_1_1_1"><i><font color="Green" title="E106">A11</font></i></a>, <a class="txt" href="finseq_6.html#E13:113_1_1_1_1_1"><i><font color="Green" title="E107">A12</font></i></a>, <a class="ref" href="finseq_6.html#T68" target="_self" title="FINSEQ_6:th.68">Th68</a></i>;<br/><b>hence </b><a NAME="E19:113_1_1_1_1_1"/>
<font color="Maroon" title="c4">x</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"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E3:113_1_1"><i><font color="Green" title="E97">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:113_1_1_1_1_2"/><i><font color="Green" title="E101">A14</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E2:113_1_1_1_1_2"/><i><font color="Green" title="E102">A15</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:113_1_1_1_1_2"/><i><font color="Green" title="E103">A16</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1_1"><i><font color="Green" title="E98">A3</font></i></a>, <a class="txt" href="finseq_6.html#E2:113_1_1_1"><i><font color="Green" title="E99">A4</font></i></a>, <a class="txt" href="finseq_6.html#E1:113_1_1_1_1_2"><i><font color="Green" title="E101">A14</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/><a NAME="E4:113_1_1_1_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1"><i><font color="Green" title="E96">A1</font></i></a>, <a class="ref" href="finseq_4.html#T31" title="FINSEQ_4:th.31">FINSEQ_4:31</a></i>;<br/><a NAME="E5:113_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="finseq_6.html#E3:113_1_1_1_1_2"><i><font color="Green" title="E103">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:113_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">p</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="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1"><i><font color="Green" title="E96">A1</font></i></a>, <a class="ref" href="finseq_6.html#T62" target="_self" title="FINSEQ_6:th.62">Th62</a></i>;<br/><a NAME="E7:113_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">x</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="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1_1"><i><font color="Green" title="E98">A3</font></i></a>, <a class="txt" href="finseq_6.html#E2:113_1_1_1_1_2"><i><font color="Green" title="E102">A15</font></i></a>, <a class="ref" href="finseq_5.html#T49" title="FINSEQ_5:th.49">FINSEQ_5:49</a></i>;<br/><a NAME="E8:113_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">x</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="c3">f</font> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1"><i><font color="Green" title="E96">A1</font></i></a>, <a class="txt" href="finseq_6.html#E3:113_1_1_1_1_2"><i><font color="Green" title="E103">A16</font></i></a>, <a class="ref" href="finseq_6.html#T65" target="_self" title="FINSEQ_6:th.65">Th65</a></i>;<br/><b>hence </b><a NAME="E9:113_1_1_1_1_2"/>
<font color="Maroon" title="c4">x</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"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E3:113_1_1"><i><font color="Green" title="E97">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:113_1_1_1_1_3"/>
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c3">f</font>
 ;<br/><div class="add"><a NAME="E2:113_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c4">x</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="c3">f</font> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1:113_1_1"><i><font color="Green" title="E96">A1</font></i></a>, <a class="txt" href="finseq_6.html#E1:113_1_1_1"><i><font color="Green" title="E98">A3</font></i></a>, <a class="txt" href="finseq_6.html#E2:113_1_1_1"><i><font color="Green" title="E99">A4</font></i></a>, <a class="ref" href="finseq_6.html#T67" target="_self" title="FINSEQ_6:th.67">Th67</a></i>;<br/><b>hence </b><a NAME="E3:113_1_1_1_1_3"/>
<font color="Maroon" title="c4">x</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"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_6.html#E3:113_1_1"><i><font color="Green" title="E97">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_2"><b>suppose </b></a><a NAME="E1:113_1_2"/>
not <font color="Maroon" title="c2">p</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="c3">f</font>
 ;<br/><div class="add"><a NAME="E2:113_1_2"/><b>then </b>
 <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="finseq_6.html#D2" target="_self" title="FINSEQ_6:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E3:113_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</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="c3">f</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T36" title="FINSEQ_5:th.36">FINSEQ_5:36</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
