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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c3">g</font> be  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green" title="E20">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 ;<br/>

<a NAME="E3:21"/><i><font color="Green" title="E21">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E4:21"/><b>then </b><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E5:21"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E23">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> )
 ;<br/>

<a NAME="E2:21_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:21_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:21"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="topreal8.html#E2:21"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="topreal8.html#E4:21"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="topreal8.html#E1:21_1"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:21"/><b>then </b><i><font color="Green" title="E23">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:21"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="topreal8.html#E3:21"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="topreal8.html#E4:21"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="finseq_6.html#T4" title="FINSEQ_6:th.4">FINSEQ_6:4</a></i>;<br/>
<a NAME="E7:21"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="ref" href="topreal8.html#T16" target="_self" title="TOPREAL8:th.16">Th16</a></i>;<br/>
<a NAME="E8:21"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E6:21"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="topreal8.html#T17" target="_self" title="TOPREAL8:th.17">Th17</a></i>;<br/>
<a NAME="E9:21"/><b>then </b><i><font color="Green" title="E25">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="topreal8.html#T13" target="_self" title="TOPREAL8:th.13">Th13</a></i>;<br/>
<a NAME="E10:21"/><i><font color="Green" title="E26">A9</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal8.html#T10" target="_self" title="TOPREAL8:th.10">Th10</a></i>;<br/>
<a NAME="E11:21"/>
<font color="Maroon" title="c3">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:21"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="revrot_1.html#T3" title="REVROT_1:th.3">REVROT_1:3</a></i>;<br/>
<b>hence </b> <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c2">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E10:21"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="finseq_6.html#D2" title="FINSEQ_6:def.2">FINSEQ_6:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="txt" href="topreal8.html#E7:21"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="topreal8.html#E9:21"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="graph_2.html#D2" title="GRAPH_2:def.2">GRAPH_2:def 2</a></i>
;<br/><br/>


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