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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">zero</a>   <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> ; <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">r</font>, <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">l9</font>, <font color="Olive" title="b4">r9</font> being    <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>  st (  for <font color="Olive" title="b5">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="Olive" title="b1">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> ) holds <br/>(  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Olive" title="b2">l</font>,<font color="Olive" title="b1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Olive" title="b3">l9</font>,<font color="Olive" title="b4">r9</font>) iff  for <font color="Olive" title="b5">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="Olive" title="b1">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b4">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b3">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">l</font>, <font color="Maroon" title="c4">l9</font>, <font color="Maroon" title="c5">r9</font> be    <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>; <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">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" title="c2">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <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> ) implies (  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c4">l9</font>,<font color="Maroon" title="c5">r9</font>) iff  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="c2">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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">l9</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="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:41"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 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" title="c2">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c4">l9</font>,<font color="Maroon" title="c5">r9</font>) iff  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="c2">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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">l9</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="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:41"/>
(  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c4">l9</font>,<font color="Maroon" title="c5">r9</font>) implies  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="c2">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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">l9</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="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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">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="c2">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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">l9</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="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) implies  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c4">l9</font>,<font color="Maroon" title="c5">r9</font>) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c4">l9</font>,<font color="Maroon" title="c5">r9</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">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="c2">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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">l9</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="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )</span><br/>

<a NAME="E2:41_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
 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" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i0</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>; <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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font></span><br/>

<span class="kw">assume </span><a NAME="E1:41_1_1"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">defpred </span><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>] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> implies ( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $2 &amp; <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $2 ) ) &amp; ( <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 implies ( <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $2 &amp; $2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 ) ) );<br/>
<a NAME="E2:41_1_1"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
 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">xi</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">xi</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_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="E1:41_1_1_1_1"/>
( ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> ) or <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font> or <font color="Maroon" title="c4">l9</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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_1_1_1_1"/>
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:41_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1"><span class="lab"><font color="Green" title="E33">A4</font></span></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="41_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/><div class="add"><a NAME="E2:41_1_1_1_1_2"/>
( <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> implies  ex <font color="Olive" title="b1">xi</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">xi</font> &amp; <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T1" target="_self" title="CHAIN_1:th.1">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:41_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A6</font></span></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="41_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:41_1_1_1_1_3"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c4">l9</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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/><div class="add"><a NAME="E2:41_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">xi</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c3">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">xi</font> &amp; <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">xi</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T2" target="_self" title="CHAIN_1:th.2">Th2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:41_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1_1_1_3"><span class="lab"><font color="Green" title="E35">A7</font></span></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>
<span class="kw">consider </span><font color="Maroon" title="c7">x</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E4:41_1_1"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
 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="c7">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:41_1_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#D3" target="_self" title="CHAIN_1:def.3">Def4</a></span>;<br/>
<a NAME="E6:41_1_1"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>;<br/>
<a NAME="E7:41_1_1"/>
( <font color="Maroon" title="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> or <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a></span>;<br/>
<a NAME="E8:41_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><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="E1:41_1_1_2"/>
(  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="c4">l9</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="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c8">x</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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) or  ex <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> st <br/>( <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; ( <font color="Maroon" title="c8">x</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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or <font color="Maroon" title="c4">l9</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="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E8:41_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="chain_1.html#T23" target="_self" title="CHAIN_1:th.23">Th24</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_1_2_1"/>
 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="c4">l9</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="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c8">x</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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:41_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:41_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_1"><span class="lab"><font color="Green" title="E35">A8</font></span></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="41_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_1_2_2"/>
 ex <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> st <br/>( <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; ( <font color="Maroon" title="c8">x</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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or <font color="Maroon" title="c4">l9</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="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:41_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_1"><span class="lab"><font color="Green" title="E35">A8</font></span></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>
<span class="kw">let </span><font color="Maroon" title="c6">i0</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>; <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="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> )</span><br/>

<div><span class="kw">hereby </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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:41_1_2"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">defpred </span><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>] <span class="kw">means </span>( <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $2 &amp; $2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> implies $2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 ) );<br/><a NAME="E2:41_1_2"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
 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">xi</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">xi</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_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="E1:41_1_2_1_1"/>
( ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c4">l9</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="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> ) or ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c2">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">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> ) or <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_1_1"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c4">l9</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="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/><div class="add"><a NAME="E2:41_1_2_1_1_1"/>
<font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">xi</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:41_1_2_1_1_1"/><span class="lab"><font color="Green" title="E36">A14</font></span>: 
<font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">xi</font>
 <span class="kw">and </span><br/><a NAME="E5:41_1_2_1_1_1"/><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<font color="Maroon" title="c8">xi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T1" target="_self" title="CHAIN_1:th.1">Th1</a></span>;<br/><a NAME="E6:41_1_2_1_1_1"/>
<font color="Maroon" title="c8">xi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_1_1"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E5:41_1_2_1_1_1"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:41_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_2_1_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E5:41_1_2_1_1_1"><span class="lab"><font color="Green" title="E37">A15</font></span></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="41_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_1_2"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c2">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">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">xi</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:41_1_2_1_1_2"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">xi</font>
 <span class="kw">and </span><br/><a NAME="E4:41_1_2_1_1_2"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
<font color="Maroon" title="c8">xi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="chain_1.html#T1" target="_self" title="CHAIN_1:th.1">Th1</a></span>;<br/><a NAME="E5:41_1_2_1_1_2"/>
<font color="Maroon" title="c8">xi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_1_2"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E4:41_1_2_1_1_2"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:41_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1_2_1_1_2"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E4:41_1_2_1_1_2"><span class="lab"><font color="Green" title="E37">A18</font></span></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="41_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:41_1_2_1_1_3"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/><div class="add"><a NAME="E2:41_1_2_1_1_3"/>
<font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/><a NAME="E3:41_1_2_1_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">xi</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">xi</font> &amp; <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T1" target="_self" title="CHAIN_1:th.1">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:41_1_2_1_1_3"/>
 ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_2_1_1_3"><span class="lab"><font color="Green" title="E35">A19</font></span></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><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E4:41_1_2"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
 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="c7">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:41_1_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#D3" target="_self" title="CHAIN_1:def.3">Def4</a></span>;<br/><a NAME="E6:41_1_2"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>;<br/><a NAME="E7:41_1_2"/>
( <font color="Maroon" title="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> or <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a></span>;<br/><a NAME="E8:41_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A22</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a></span>;<br/><a NAME="E9:41_1_2"/>
(  not <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> or  not <font color="Maroon" title="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:41_1_2"/><span class="kw">then </span>
 ex <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> st <br/>( <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; ( <font color="Maroon" title="c8">x</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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or <font color="Maroon" title="c4">l9</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="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E8:41_1_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="chain_1.html#T23" target="_self" title="CHAIN_1:th.23">Th24</a></span>;<br/><span class="kw">hence </span><a NAME="E11:41_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_2"><span class="lab"><font color="Green" title="E35">A20</font></span></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>

<span class="kw">thus </span><a NAME="E4:41_1"/>
<font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a></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">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font></span><br/>

<div><span class="kw">hereby </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 class="add"><span class="kw">assume </span><a NAME="E1:41_1_3"/><span class="lab"><font color="Green" title="E33">A23</font></span>: 
<font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">defpred </span><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>] <span class="kw">means </span>( <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> $2 &amp; $2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> implies $2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 ) );<br/><a NAME="E2:41_1_3"/><span class="lab"><font color="Green" title="E34">A24</font></span>: 
 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">xi</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">xi</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_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="E1:41_1_3_1_1"/>
( ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c5">r9</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#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> ) or ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c3">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#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> ) or <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_3_1_1_1"/><span class="lab"><font color="Green" title="E35">A25</font></span>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c5">r9</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#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/><div class="add"><a NAME="E2:41_1_3_1_1_1"/>
<font color="Maroon" title="c4">l9</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#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">xi</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:41_1_3_1_1_1"/><span class="lab"><font color="Green" title="E36">A26</font></span>: 
<font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">xi</font>
 <span class="kw">and </span><br/><a NAME="E5:41_1_3_1_1_1"/><span class="lab"><font color="Green" title="E37">A27</font></span>: 
<font color="Maroon" title="c8">xi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T1" target="_self" title="CHAIN_1:th.1">Th1</a></span>;<br/><a NAME="E6:41_1_3_1_1_1"/>
<font color="Maroon" title="c8">xi</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_3_1_1_1"><span class="lab"><font color="Green" title="E35">A25</font></span></a>, <a class="txt" href="#E4:41_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:41_1_3_1_1_1"/>
 ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_3_1_1_1"><span class="lab"><font color="Green" title="E36">A26</font></span></a>, <a class="txt" href="#E5:41_1_3_1_1_1"><span class="lab"><font color="Green" title="E37">A27</font></span></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="41_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_3_1_1_2"/><span class="lab"><font color="Green" title="E35">A28</font></span>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c3">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#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">xi</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:41_1_3_1_1_2"/><span class="lab"><font color="Green" title="E36">A29</font></span>: 
<font color="Maroon" title="c3">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">xi</font>
 <span class="kw">and </span><br/><a NAME="E4:41_1_3_1_1_2"/><span class="lab"><font color="Green" title="E37">A30</font></span>: 
<font color="Maroon" title="c8">xi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_3"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="ref" href="chain_1.html#T1" target="_self" title="CHAIN_1:th.1">Th1</a></span>;<br/><a NAME="E5:41_1_3_1_1_2"/>
<font color="Maroon" title="c8">xi</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_3_1_1_2"><span class="lab"><font color="Green" title="E35">A28</font></span></a>, <a class="txt" href="#E3:41_1_3_1_1_2"><span class="lab"><font color="Green" title="E36">A29</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:41_1_3_1_1_2"/>
 ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1_3_1_1_2"><span class="lab"><font color="Green" title="E36">A29</font></span></a>, <a class="txt" href="#E4:41_1_3_1_1_2"><span class="lab"><font color="Green" title="E37">A30</font></span></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="41_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:41_1_3_1_1_3"/><span class="lab"><font color="Green" title="E35">A31</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">i0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]</span><br/><div class="add"><a NAME="E2:41_1_3_1_1_3"/>
<font color="Maroon" title="c4">l9</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#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/><a NAME="E3:41_1_3_1_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">xi</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">xi</font> &amp; <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#T1" target="_self" title="CHAIN_1:th.1">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:41_1_3_1_1_3"/>
 ex <font color="Olive" title="b1">xi</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="Maroon" title="c7">i</font>,<font color="Olive" title="b1">xi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_3_1_1_3"><span class="lab"><font color="Green" title="E35">A31</font></span></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><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E4:41_1_3"/><span class="lab"><font color="Green" title="E35">A32</font></span>: 
 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="c7">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:41_1_3"><span class="lab"><font color="Green" title="E34">A24</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="chain_1.html#D3" target="_self" title="CHAIN_1:def.3">Def4</a></span>;<br/><a NAME="E6:41_1_3"/><span class="lab"><font color="Green" title="E36">A33</font></span>: 
<font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>;<br/><a NAME="E7:41_1_3"/>
( <font color="Maroon" title="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> or <font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_3"><span class="lab"><font color="Green" title="E35">A32</font></span></a></span>;<br/><a NAME="E8:41_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A34</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_1_3"><span class="lab"><font color="Green" title="E36">A33</font></span></a></span>;<br/><a NAME="E9:41_1_3"/>
(  not <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> or  not <font color="Maroon" title="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">i0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:41_1_3"/><span class="kw">then </span>
 ex <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> st <br/>( <font color="Maroon" title="c4">l9</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#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; ( <font color="Maroon" title="c8">x</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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> or <font color="Maroon" title="c4">l9</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="c8">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E8:41_1_3"><span class="lab"><font color="Green" title="E37">A34</font></span></a>, <a class="ref" href="chain_1.html#T23" target="_self" title="CHAIN_1:th.23">Th24</a></span>;<br/><span class="kw">hence </span><a NAME="E11:41_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_3"><span class="lab"><font color="Green" title="E35">A32</font></span></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>

<span class="kw">assume </span><a NAME="E3:41"/><span class="lab"><font color="Green" title="E31">A35</font></span>: 
 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="c2">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="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">r9</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">l9</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="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c4">l9</font>,<font color="Maroon" title="c5">r9</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">r</font>) or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c4">l9</font>,<font color="Maroon" title="c5">r9</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E4:41"/><span class="lab"><font color="Green" title="E32">A36</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">r</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c4">l9</font>,<font color="Maroon" title="c5">r9</font>)</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</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> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">i0</font> =  the    <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="E6:41"/><span class="lab"><font color="Green" title="E33">A37</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E31">A35</font></span></a></span>;<br/>
<a NAME="E7:41"/>
<font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E31">A35</font></span></a></span>;<br/>
<a NAME="E8:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A38</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E33">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:41"/>
<font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E31">A35</font></span></a></span>;<br/>
<a NAME="E10:41"/><span class="kw">then </span>
<font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:41"><span class="lab"><font color="Green" title="E34">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:41"/><span class="kw">then </span>
( <font color="Maroon" title="c7">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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> or <font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a>  the    <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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><span class="kw"> such that </span><br/><a NAME="E13:41"/>
<font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E14:41"/><span class="lab"><font color="Green" title="E35">A39</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> or <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E32">A36</font></span></a>, <a class="ref" href="chain_1.html#T23" target="_self" title="CHAIN_1:th.23">Th24</a></span>;<br/>
<a NAME="E15:41"/><span class="lab"><font color="Green" title="E36">A40</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E31">A35</font></span></a></span>;<br/>
<a NAME="E16:41"/><span class="lab"><font color="Green" title="E37">A41</font></span>: 
<font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E31">A35</font></span></a></span>;<br/>
<a NAME="E17:41"/><span class="lab"><font color="Green" title="E38">A42</font></span>: 
<font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E31">A35</font></span></a></span>;<br/>
<a NAME="E18:41"/>
( <font color="Maroon" title="c7">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> or <font color="Maroon" title="c4">l9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:41"><span class="lab"><font color="Green" title="E35">A39</font></span></a>, <a class="txt" href="#E15:41"><span class="lab"><font color="Green" title="E36">A40</font></span></a>, <a class="txt" href="#E16:41"><span class="lab"><font color="Green" title="E37">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:41"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> (<font color="Maroon" title="c4">l9</font>,<font color="Maroon" title="c5">r9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:41"><span class="lab"><font color="Green" title="E38">A42</font></span></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>
