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<b>let </b><font color="Maroon" title="c1">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <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>assume </b><b>that </b><br/><a NAME="E1:2"/><i><font color="Green" title="E3">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 <b>and </b><br/><a NAME="E2:2"/><i><font color="Green" title="E4">A2</font></i>: 
 ex <font color="Olive" title="b1">r</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>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E3:2"/><i><font color="Green" title="E5">A3</font></i>: 
for <font color="Olive" title="b1">r</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>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E4:2"/><i><font color="Green" title="E6">A4</font></i>: 
 ex <font color="Olive" title="b1">g</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>  st <br/>for <font color="Olive" title="b2">r</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>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>not <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">g1</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="E6:2"/><i><font color="Green" title="E7">A5</font></i>: 
for <font color="Olive" title="b1">r</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>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>not <font color="Maroon" title="c3">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E4:2"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <b>means </b>for <font color="Olive" title="b1">r</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>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>not $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font>;<br/>
<b>consider </b><font color="Maroon" title="c4">Y</font> being   <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> <b> such that </b><br/><a NAME="E8:2"/><i><font color="Green" title="E8">A6</font></i>: 
for <font color="Olive" title="b1">p1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p1</font>] )
 <b>from </b><i><a class="ref" href="real_1.html#S1" title="REAL_1:sch.1">REAL_1:sch 1</a>();<br/></i>
<a NAME="E9:2"/>
<font color="Maroon" title="c3">g1</font> is   <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/>
<a NAME="E10:2"/><b>then </b><i><font color="Green" title="E9">A7</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E6:2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="seq_4.html#E8:2"><i><font color="Green" title="E8">A6</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">p1</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:2_1"/><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E2:2_1"/><i><font color="Green" title="E11">A9</font></i>: 
<font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ;<br/><a NAME="E3:2_1"/><i><font color="Green" title="E12">A10</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E3:2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="seq_4.html#E1:2_1"><i><font color="Green" title="E10">A8</font></i></a></i>;<br/><a NAME="E4:2_1"/><i><font color="Green" title="E13">A11</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E5:2_1"/>
<font color="Maroon" title="c5">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">p1</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E8:2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="seq_4.html#E2:2_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="seq_4.html#E3:2_1"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E6:2_1"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p1</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E4:2_1"><i><font color="Green" title="E13">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c5">g2</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="E13:2"/><i><font color="Green" title="E10">A12</font></i>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">p1</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>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> holds <br/>( <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Maroon" title="c5">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">p1</font> )
 <b>by </b><i><a class="ref" href="seq_4.html#T8" target="_self" title="SEQ_4:th.8">Th8</a></i>;<br/>
<a NAME="E14:2"/><i><font color="Green" title="E11">A13</font></i>: 
<font color="Maroon" title="c5">g2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">p</font> is   <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/>
<div><i><font color="Green" title="E12">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2"/>
not <font color="Maroon" title="c5">g2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">r1</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="E3:2_2"/><i><font color="Green" title="E13">A15</font></i>: 
<font color="Maroon" title="c6">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E4:2_2"/><i><font color="Green" title="E14">A16</font></i>: 
<font color="Maroon" title="c5">g2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r1</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E8:2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="seq_4.html#E14:2"><i><font color="Green" title="E11">A13</font></i></a></i>;<br/><a NAME="E5:2_2"/><i><font color="Green" title="E15">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E4:2_2"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:2_2"/>
<font color="Maroon" title="c6">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E3:2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="seq_4.html#E3:2_2"><i><font color="Green" title="E13">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E7:2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_4.html#E10:2"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="seq_4.html#E13:2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="seq_4.html#E5:2_2"><i><font color="Green" title="E15">A17</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:2"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="seq_4.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E17:2"/><b>then </b>
<font color="Maroon" title="c5">g2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">g2</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E18:2"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_4.html#E2:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="seq_4.html#E13:2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="seq_4.html#E15:2"><i><font color="Green" title="E12">A14</font></i></a></i>;<br/>


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