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<b>let </b><font color="Maroon" title="c1">d'</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being 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> <br/> for <font color="Olive" title="b2">l</font>, <font color="Olive" title="b3">r</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="Olive" title="b1">d</font><br/> for <font color="Olive" title="b4">G</font> being   <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Olive" title="b1">d</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">d'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 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="b3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c1">d'</font>,<font color="Olive" title="b4">G</font> iff  ex <font color="Olive" title="b5">i0</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="Olive" title="b1">d</font> st <br/>( <font color="Olive" title="b2">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i0</font> &amp; <font color="Olive" title="b2">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b5">i0</font> &amp; ( for <font color="Olive" title="b6">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="Olive" title="b1">d</font>  st <font color="Olive" title="b6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">i0</font> holds <br/>( <font color="Olive" title="b2">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Olive" title="b4">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b6">i</font> ) ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">l</font>, <font color="Olive" title="b2">r</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="c2">d</font><br/> for <font color="Olive" title="b3">G</font> being   <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Maroon" title="c2">d</font>  st <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">d'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/>(  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Olive" title="b1">l</font>,<font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c1">d'</font>,<font color="Olive" title="b3">G</font> iff  ex <font color="Olive" title="b4">i0</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="c2">d</font> st <br/>( <font color="Olive" title="b1">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">i0</font> &amp; <font color="Olive" title="b1">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b4">i0</font> &amp; ( 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="c2">d</font>  st <font color="Olive" title="b5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">i0</font> holds <br/>( <font color="Olive" title="b1">l</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">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Olive" title="b3">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b5">i</font> ) ) ) )</span><br/><b>let </b><font color="Maroon" title="c3">l</font>, <font color="Maroon" title="c4">r</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="c2">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Maroon" title="c2">d</font>  st <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">d'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/>(  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c1">d'</font>,<font color="Olive" title="b1">G</font> iff  ex <font color="Olive" title="b2">i0</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="c2">d</font> st <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i0</font> &amp; <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b2">i0</font> &amp; ( for <font color="Olive" title="b3">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="c2">d</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i0</font> holds <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Olive" title="b1">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b3">i</font> ) ) ) )</span><br/><b>let </b><font color="Maroon" title="c5">G</font> be    <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Maroon" title="c2">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">d'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 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="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c1">d'</font>,<font color="Maroon" title="c5">G</font> iff  ex <font color="Olive" title="b1">i0</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="c2">d</font> st <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> &amp; <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i0</font> &amp; ( for <font color="Olive" title="b2">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="c2">d</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i0</font> holds <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b2">i</font> ) ) ) ) )</span><br/>









<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">d'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c1">d'</font>,<font color="Maroon" title="c5">G</font> iff  ex <font color="Olive" title="b1">i0</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="c2">d</font> st <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> &amp; <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i0</font> &amp; ( for <font color="Olive" title="b2">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="c2">d</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i0</font> holds <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b2">i</font> ) ) ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">i0</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="c2">d</font> st <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> &amp; <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i0</font> &amp; ( for <font color="Olive" title="b2">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="c2">d</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i0</font> holds <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b2">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="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c1">d'</font>,<font color="Maroon" title="c5">G</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:58_1"/>
 <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c1">d'</font>,<font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i0</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="c2">d</font> st <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> &amp; <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i0</font> &amp; ( for <font color="Olive" title="b2">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="c2">d</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i0</font> holds <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b2">i</font> ) ) )</span><br/><b>then consider </b><font color="Maroon" title="c6">l'</font>, <font color="Maroon" title="c7">r'</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="c2">d</font>, <font color="Maroon" title="c8">i0</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="c2">d</font><b> such that </b><br/><a NAME="E3:58_1"/><i><font color="Green" title="E45">A2</font></i>: 
(  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K3" title="CHAIN_1:func.3">cell</a> <font color="Maroon" title="c6">l'</font>,<font color="Maroon" title="c7">r'</font> &amp; <font color="Maroon" title="c6">l'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i0</font> &amp; <font color="Maroon" title="c6">l'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c8">i0</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">d</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">i0</font> holds <br/>( <font color="Maroon" title="c6">l'</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="c7">r'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">l'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:58"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="chain_1.html#T42" target="_self" title="CHAIN_1:th.42">Th42</a></i>;<br/><b>take </b><font color="Maroon" title="c9">i0</font> = <font color="Maroon" title="c8">i0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c9">i0</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">d</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i0</font> holds <br/>( <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 href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) ) )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">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="c2">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">l'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span><br/><a NAME="E1:58_1_1"/>
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i0</font> or <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i0</font> )
 ;<br/><b>hence </b><a NAME="E2:58_1_1"/>
<font color="Maroon" title="c6">l'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r'</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:58_1"><i><font color="Green" title="E45">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:58_1"/><b>then </b>
( <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l'</font> &amp; <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r'</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:58_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="chain_1.html#T32" target="_self" title="CHAIN_1:th.32">Th32</a></i>;<br/><b>hence </b><a NAME="E6:58_1"/>
( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c9">i0</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">d</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i0</font> holds <br/>( <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 href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:58_1"><i><font color="Green" title="E45">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E3:58"/>
(  ex <font color="Olive" title="b1">i0</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="c2">d</font> st <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> &amp; <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i0</font> &amp; ( for <font color="Olive" title="b2">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="c2">d</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i0</font> holds <br/>( <font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c5">G</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b2">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="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c1">d'</font>,<font color="Maroon" title="c5">G</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:58"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="chain_1.html#T42" target="_self" title="CHAIN_1:th.42">Th42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
