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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> ; <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">li</font>, <font color="Olive" title="b2">ri</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">li</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; <font color="Olive" title="b2">ri</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; <font color="Olive" title="b2">ri</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">li</font> &amp; ( for <font color="Olive" title="b3">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b3">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="Olive" title="b3">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">ri</font> &amp; not <font color="Olive" title="b1">li</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">xi</font> ) ) )</span><br/>

<b>consider </b><font color="Maroon" title="c2">li</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E2:29"/><i><font color="Green" title="E18">A1</font></i>: 
( <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; ( 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/><font color="Maroon" title="c2">li</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">xi</font> ) )
 <b>by </b><i><a class="ref" href="chain_1.html#T12" target="_self" title="CHAIN_1:th.12">Th12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">ri</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:29"/><i><font color="Green" title="E19">A2</font></i>: 
( <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; ( 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/><font color="Maroon" title="c3">ri</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">xi</font> ) )
 <b>by </b><i><a class="ref" href="chain_1.html#T13" target="_self" title="CHAIN_1:th.13">Th13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">li</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">ri</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <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="Olive" title="b1">ri</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Gi</font> &amp; <font color="Olive" title="b1">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="b2">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">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="Olive" title="b2">xi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">ri</font> &amp; 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="b2">xi</font> ) ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c3">ri</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">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="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="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> &amp; 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> ) ) )</span><br/>

<a NAME="E5:29"/><i><font color="Green" title="E20">A3</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="chain_1.html#E2:29"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="chain_1.html#E4:29"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c2">li</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ri</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green" title="E22">A5</font></i>: 
( <font color="Maroon" title="c4">x1</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">x2</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">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x2</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#D3" target="_self" title="CHAIN_1:def.3">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c7">x2</font> = <font color="Maroon" title="c5">x2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="chain_1.html#E3:29_1"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/><a NAME="E5:29_1"/>
( <font color="Maroon" title="c3">ri</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">li</font> &amp; <font color="Maroon" title="c3">ri</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">li</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:29"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="chain_1.html#E4:29"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="chain_1.html#E3:29_1"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/><a NAME="E6:29_1"/><b>then </b>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">li</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">li</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:29_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E7:29_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chain_1.html#E3:29_1"><i><font color="Green" title="E22">A5</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>hence </b><a NAME="E7:29"/>
( <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="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="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> &amp; 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> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:29"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="chain_1.html#E4:29"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="chain_1.html#E5:29"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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