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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</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"> ( <font color="Maroon" title="c1">A</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c2">B</font> implies  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<b>consider </b><font color="Maroon" title="c3">r0</font>, <font color="Maroon" title="c4">s0</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:15"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c3">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E4:15"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c4">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E5:15"/><i><font color="Green" title="E16">A4</font></i>: 
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">s0</font></span>)</span>
 <b>by </b><i><a class="ref" href="jct_misc.html#T18" target="_self" title="JCT_MISC:th.18">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">r0</font> = <font color="Maroon" title="c3">r0</font>, <font color="Maroon" title="c6">s0</font> = <font color="Maroon" title="c4">s0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E7:15"/>
 <a href="jct_misc.html#K1" title="JCT_MISC:func.1">dist</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E8:15"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">s0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E5:15"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="jct_misc.html#T19" target="_self" title="JCT_MISC:th.19">Th19</a></i>;<br/>
<a NAME="E9:15"/><b>then </b>
<font color="Maroon" title="c5">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s0</font>
 
<b>by </b><i><a class="ref" href="goboard7.html#T2" title="GOBOARD7:th.2">GOBOARD7:2</a></i>;<br/>
<b>hence </b><a NAME="E10:15"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E3:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E4:15"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</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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