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

<b>let </b><font color="Maroon" title="c1">i1</font>, <font color="Maroon" title="c2">i2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon" title="c4">g1</font>, <font color="Maroon" title="c5">g2</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>









<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E53">A1</font></i>: 
( <font color="Maroon" title="c4">g1</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> &amp; <font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_1"/>
( <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font> or <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">i2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1"><b>case </b></a><a NAME="E1:57_1_1_1"/><i><font color="Green" title="E54">A2</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><a NAME="E2:57_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:57"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="jordan4.html#T37" target="_self" title="JORDAN4:th.37">Th37</a></i>;<br/><b>hence </b><a NAME="E3:57_1_1_1"/>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:57"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="jordan4.html#E1:57_1_1_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="jordan4.html#T37" target="_self" title="JORDAN4:th.37">Th37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_2"><b>case </b></a><a NAME="E1:57_1_1_2"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><a NAME="E2:57_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,1,<font color="Maroon" title="c2">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:57"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="jordan4.html#T38" target="_self" title="JORDAN4:th.38">Th38</a></i>;<br/><b>hence </b><a NAME="E3:57_1_1_2"/>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:57"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="jordan4.html#E1:57_1_1_2"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="jordan4.html#T38" target="_self" title="JORDAN4:th.38">Th38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:57"/>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font>
 ;<br/>


</div>
