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<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</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="b2">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> holds   <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">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="Olive" title="b2">f</font>,<font color="Olive" title="b1">p</font>)</span>)</span></span><br/><span class="kw">let </span><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>; <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">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> holds   <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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="Olive" title="b1">f</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#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="c3">f</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_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="E1:111_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="111_1_1"><span class="kw">suppose </span></a><a NAME="E1:111_1_1"/><span class="lab"><font color="Green" title="E94">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#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="c3">f</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:111_1_1"/><span class="kw">then </span>
 <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#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="E3:111_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A2</font></span>: 
 <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#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>;<br/><span class="kw">thus </span><a NAME="E4:111_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#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="c3">f</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:111_1_1_1"/><span class="lab"><font color="Green" title="E96">A3</font></span>: 
<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#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 ; <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="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></span><br/>

<a NAME="E2:111_1_1_1"/><span class="lab"><font color="Green" title="E97">A4</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="c3">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> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T33" title="FINSEQ_5:th.33">FINSEQ_5:33</a></span>;<br/>
<a NAME="E3:111_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1_1"><span class="lab"><font color="Green" title="E96">A3</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_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="E1:111_1_1_1_1"/>
( ( <font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</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#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> ) or ( <font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</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#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> ) or <font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:111_1_1_1_1_1"/><span class="lab"><font color="Green" title="E99">A6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E2:111_1_1_1_1_1"/><span class="lab"><font color="Green" title="E100">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</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#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 ; <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="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></span><br/><div class="add"><a NAME="E3:111_1_1_1_1_1"/><span class="lab"><font color="Green" title="E101">A8</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="ref" href="finseq_4.html#T21" title="FINSEQ_4:th.21">FINSEQ_4:21</a></span>;<br/><a NAME="E4:111_1_1_1_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font> <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="#E1:111_1_1_1"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E2:111_1_1_1"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="txt" href="#E1:111_1_1_1_1_1"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="ref" href="finseq_4.html#T21" title="FINSEQ_4:th.21">FINSEQ_4:21</a></span>;<br/><a NAME="E5:111_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_1_1_1_1"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:111_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></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="#E1:111_1_1"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="ref" href="finseq_6.html#T56" target="_self" title="FINSEQ_6:th.56">Th61</a></span>;<br/><a NAME="E7:111_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E8:111_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_1_1"><span class="lab"><font color="Green" title="E98">A5</font></span></a></span>;<br/><a NAME="E10:111_1_1_1_1_1"/><span class="lab"><font color="Green" title="E103">A10</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</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#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1_1"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="ref" href="finseq_4.html#T20" title="FINSEQ_4:th.20">FINSEQ_4:20</a></span>;<br/><a NAME="E11:111_1_1_1_1_1"/>
<font color="Maroon" title="c3">f</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="#E1:111_1_1"><span class="lab"><font color="Green" title="E94">A1</font></span></a></span>;<br/><a NAME="E12:111_1_1_1_1_1"/><span class="kw">then </span>
 <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E13:111_1_1_1_1_1"/><span class="kw">then </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="real_1.html#K9" title="REAL_1:func.9">-</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#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></span>;<br/><a NAME="E14:111_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A11</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> <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#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E15:111_1_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</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#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1_1"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="ref" href="finseq_4.html#T21" title="FINSEQ_4:th.21">FINSEQ_4:21</a></span>;<br/><a NAME="E16:111_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <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>  <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="#E14:111_1_1_1_1_1"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<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#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <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="#E1:111_1_1_1"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="ref" href="finseq_5.html#T38" title="FINSEQ_5:th.38">FINSEQ_5:38</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <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="#E10:111_1_1_1_1_1"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>
;<br/><a NAME="E18:111_1_1_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E16:111_1_1_1_1_1"><span class="lab"><font color="Green" title="E105">A12</font></span></a>, <a class="txt" href="#E7:111_1_1_1_1_1"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="ref" href="finseq_6.html#T63" target="_self" title="FINSEQ_6:th.63">Th68</a></span>;<br/><span class="kw">hence </span><a NAME="E19:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_1"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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="111_1_1_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:111_1_1_1_1_2"/><span class="lab"><font color="Green" title="E99">A13</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E2:111_1_1_1_1_2"/><span class="lab"><font color="Green" title="E100">A14</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</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#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 ; <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="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></span><br/><div class="add"><a NAME="E3:111_1_1_1_1_2"/><span class="lab"><font color="Green" title="E101">A15</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font> <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="#E1:111_1_1_1"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E2:111_1_1_1"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="txt" href="#E1:111_1_1_1_1_2"><span class="lab"><font color="Green" title="E99">A13</font></span></a>, <a class="ref" href="finseq_4.html#T21" title="FINSEQ_4:th.21">FINSEQ_4:21</a></span>;<br/><a NAME="E4:111_1_1_1_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="ref" href="finseq_4.html#T21" title="FINSEQ_4:th.21">FINSEQ_4:21</a></span>;<br/><a NAME="E5:111_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_1_1_1_2"><span class="lab"><font color="Green" title="E101">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:111_1_1_1_1_2"/><span class="kw">then </span>
<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#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="ref" href="finseq_6.html#T57" target="_self" title="FINSEQ_6:th.57">Th62</a></span>;<br/><a NAME="E7:111_1_1_1_1_2"/><span class="kw">then </span>
<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#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">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1_1"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E2:111_1_1_1_1_2"><span class="lab"><font color="Green" title="E100">A14</font></span></a>, <a class="ref" href="finseq_5.html#T46" title="FINSEQ_5:th.46">FINSEQ_5:46</a></span>;<br/><a NAME="E8:111_1_1_1_1_2"/><span class="kw">then </span>
<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#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E3:111_1_1_1_1_2"><span class="lab"><font color="Green" title="E101">A15</font></span></a>, <a class="ref" href="finseq_6.html#T60" target="_self" title="FINSEQ_6:th.60">Th65</a></span>;<br/><span class="kw">hence </span><a NAME="E9:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_1"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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="111_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:111_1_1_1_1_3"/>
<font color="Maroon" title="c4">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c3">f</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="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></span><br/><div class="add"><a NAME="E2:111_1_1_1_1_3"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E1:111_1_1_1"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E2:111_1_1_1"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="ref" href="finseq_6.html#T62" target="_self" title="FINSEQ_6:th.62">Th67</a></span>;<br/><span class="kw">hence </span><a NAME="E3:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_1"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_2"><span class="kw">suppose </span></a><a NAME="E1:111_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#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="c3">f</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:111_1_2"/><span class="kw">then </span>
 <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>
 <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/><span class="kw">hence </span><a NAME="E3:111_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#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="c3">f</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T33" title="FINSEQ_5:th.33">FINSEQ_5:33</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>
