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<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="membered.html#V2" title="MEMBERED:attr.2">real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green" title="E16">A1</font></i>: 
not <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">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="E4:11"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c2">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:11"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="membered.html#T7" title="MEMBERED:th.7">MEMBERED:7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E6:11"/><i><font color="Green" title="E19">A4</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="c1">X</font> holds <br/><font color="Maroon" title="c3">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:11"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="seq_4.html#D2" target="_self" title="SEQ_4:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c1">X</font> as   <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>by </b><i><a class="ref" href="membered.html#T2" title="MEMBERED:th.2">MEMBERED:2</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="c4">X</font> holds <br/>$1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">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="E9:11"/><i><font color="Green" title="E20">A5</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</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="E10:11"/>
<font color="Maroon" title="c3">p1</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="E11:11"/><b>then </b><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E6:11"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="seq_4.html#E9:11"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>
<a NAME="E12:11"/>
for <font color="Olive" title="b1">p</font>, <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="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E9:11"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">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="E14:11"/><i><font color="Green" title="E22">A7</font></i>: 
for <font color="Olive" title="b1">p</font>, <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="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r</font> )
 <b>by </b><i><a class="ref" href="axioms.html#T26" title="AXIOMS:th.26">AXIOMS:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">g1</font> = <font color="Maroon" title="c6">g1</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>take </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">g1</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c9">s1</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><a NAME="E1:11_1"/><i><font color="Green" title="E23">A8</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s1</font>
 <b>and </b><br/><a NAME="E2:11_1"/><i><font color="Green" title="E24">A9</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="c4">X</font> holds <br/>not <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">g</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">s1</font>
 ;<br/><a NAME="E3:11_1"/>
<font color="Maroon" title="c8">g</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">s1</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="E4:11_1"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E9:11"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="seq_4.html#E2:11_1"><i><font color="Green" title="E24">A9</font></i></a></i>;<br/><a NAME="E5:11_1"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">g</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E4:11"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="seq_4.html#E14:11"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/><a NAME="E6:11_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">g</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">g</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E7:11_1"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_4.html#E1:11_1"><i><font color="Green" title="E23">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:11"/>
( ( 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="c1">X</font> holds <br/><font color="Maroon" title="c8">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> ) &amp; ( for <font color="Olive" title="b1">s</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 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> ex <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 <br/>( <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">g</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">s</font> ) ) )
 <b>by </b><i><a class="txt" href="seq_4.html#E11:11"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="seq_4.html#E14:11"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/>


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