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<b>let </b><font color="Maroon" title="c1">Gi</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">li</font>, <font color="Maroon" title="c3">ri</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<b>thus </b><a NAME="E1:32"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">li</font>,<font color="Maroon" title="c3">ri</font></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="c1">Gi</font> implies ( <font color="Maroon" title="c2">li</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; <font color="Maroon" title="c3">ri</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; ( ( <font color="Maroon" title="c2">li</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">ri</font> &amp; ( for <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; <font color="Maroon" title="c2">li</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">xi</font> holds <br/> not <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">ri</font> ) ) or ( <font color="Maroon" title="c3">ri</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">li</font> &amp; ( for <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> holds <br/>( not <font color="Maroon" title="c2">li</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">xi</font> &amp; not <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">ri</font> ) ) ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">li</font>,<font color="Maroon" title="c3">ri</font></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="c1">Gi</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">li'</font>, <font color="Maroon" title="c5">ri'</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green" title="E21">A1</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">li</font>,<font color="Maroon" title="c3">ri</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">li'</font>,<font color="Maroon" title="c5">ri'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c4">li'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; <font color="Maroon" title="c5">ri'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; ( ( <font color="Maroon" title="c4">li'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">ri'</font> &amp; ( for <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; <font color="Maroon" title="c4">li'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">xi</font> holds <br/> not <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">ri'</font> ) ) or ( <font color="Maroon" title="c5">ri'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">li'</font> &amp; ( for <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> holds <br/>( not <font color="Maroon" title="c4">li'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">xi</font> &amp; not <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">ri'</font> ) ) ) ) )
 <b>by </b><i><a class="ref" href="chain_1.html#D6" target="_self" title="CHAIN_1:def.6">Def6</a></i>;<br/>
<a NAME="E4:32_1"/>
( <font color="Maroon" title="c4">li'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">li</font> &amp; <font color="Maroon" title="c5">ri'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ri</font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E3:32_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:32_1"/>
( <font color="Maroon" title="c2">li</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; <font color="Maroon" title="c3">ri</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; ( ( <font color="Maroon" title="c2">li</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">ri</font> &amp; ( for <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; <font color="Maroon" title="c2">li</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">xi</font> holds <br/> not <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">ri</font> ) ) or ( <font color="Maroon" title="c3">ri</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">li</font> &amp; ( for <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> holds <br/>( not <font color="Maroon" title="c2">li</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">xi</font> &amp; not <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">ri</font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:32_1"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:32"/>
( <font color="Maroon" title="c2">li</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; <font color="Maroon" title="c3">ri</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; ( ( <font color="Maroon" title="c2">li</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">ri</font> &amp; ( for <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; <font color="Maroon" title="c2">li</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">xi</font> holds <br/> not <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">ri</font> ) ) or ( <font color="Maroon" title="c3">ri</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">li</font> &amp; ( for <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> holds <br/>( not <font color="Maroon" title="c2">li</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">xi</font> &amp; not <font color="Olive" title="b1">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">ri</font> ) ) ) ) implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">li</font>,<font color="Maroon" title="c3">ri</font></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="c1">Gi</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#D6" target="_self" title="CHAIN_1:def.6">Def6</a></i>;<br/>


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