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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <b>means </b>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> $1 &amp; ( for <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> $1 holds <br/><font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $2 ) );<br/>
<a NAME="E1:70_1_1"/><i><font color="Green" title="E48">A1</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="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>  ex <font color="Olive" title="b2">li</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">li</font>]
 
<b>by </b><i><a class="ref" href="chain_1.html#T12" target="_self" title="CHAIN_1:th.12">Th12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">l</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:70_1_1"/><i><font color="Green" title="E49">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="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="chain_1.html#E1:70_1_1"><i><font color="Green" title="E48">A1</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">l</font> = <font color="Maroon" title="c3">l</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">d</font> <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self" title="CHAIN_1:def.4">Def4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <b>means </b>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> $1 &amp; ( for <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> $1 holds <br/><font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> $2 ) );<br/>
<a NAME="E5:70_1_1"/><i><font color="Green" title="E50">A3</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="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>  ex <font color="Olive" title="b2">ri</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">ri</font>]
 
<b>by </b><i><a class="ref" href="chain_1.html#T13" target="_self" title="CHAIN_1:th.13">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">r</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E7:70_1_1"/><i><font color="Green" title="E51">A4</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="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="chain_1.html#E5:70_1_1"><i><font color="Green" title="E50">A3</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c5">r</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">d</font> <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self" title="CHAIN_1:def.4">Def4</a></i>;<br/>
<div><i><font color="Green" title="E52">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>;<br/><a NAME="E1:70_1_1_1"/>
( <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:70_1_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="chain_1.html#E7:70_1_1"><i><font color="Green" title="E51">A4</font></i></a></i>;<br/><a NAME="E2:70_1_1_1"/><b>then </b><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:70_1_1"><i><font color="Green" title="E49">A2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_1_1_1"/><i><font color="Green" title="E54">A7</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 ;<br/><b>consider </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:70_1_1_1_1"/><i><font color="Green" title="E55">A8</font></i>: 
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c8">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">x2</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#D3" target="_self" title="CHAIN_1:def.3">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c11">x2</font> = <font color="Maroon" title="c9">x2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="chain_1.html#E3:70_1_1_1_1"><i><font color="Green" title="E55">A8</font></i></a></i>;<br/><a NAME="E5:70_1_1_1_1"/>
( <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">x1</font> &amp; <font color="Maroon" title="c10">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">x2</font> &amp; <font color="Maroon" title="c11">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:70_1_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="chain_1.html#E7:70_1_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="chain_1.html#E3:70_1_1_1_1"><i><font color="Green" title="E55">A8</font></i></a></i>;<br/><a NAME="E6:70_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:70_1_1_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E7:70_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chain_1.html#E3:70_1_1_1_1"><i><font color="Green" title="E55">A8</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:70_1_1_1"/>
<font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E2:70_1_1_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E53">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font>;<br/><a NAME="E1:70_1_1_2"/>
( <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; ( for <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i</font> holds <br/>( not <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">xi</font> &amp; not <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:70_1_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="chain_1.html#E7:70_1_1"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="chain_1.html#E9:70_1_1"><i><font color="Green" title="E52">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E2:70_1_1_2"/>
( <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c7">i</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T17" target="_self" title="CHAIN_1:th.17">Th17</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">A</font> =  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c4">l</font>,<font color="Maroon" title="c6">r</font> as   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon" title="c1">d</font>,<font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="chain_1.html#T34" target="_self" title="CHAIN_1:th.34">Th34</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">A</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c4">l</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">r</font>
;<br/>

<b>thus </b><a NAME="E12:70_1_1"/>
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c4">l</font>,<font color="Maroon" title="c6">r</font> &amp; ( 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="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font> holds <br/> ( <font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c2">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E10:70_1_1"><i><font color="Green" title="E53">A9</font></i></a></i>;<br/>


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