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<b>let </b><font color="Maroon" title="c1">s</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">s</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> </span>}</span> </span><br/>

<a NAME="E1:237"/><i><font color="Green" title="E98">F</font></i>: 
<font color="Maroon" title="c1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.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>thus </b><a NAME="E2:237"/>
<span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">s</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">s</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="237_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="membered.html#D9" title="MEMBERED:def.9">MEMBERED:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">s</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:237_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">s</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> </span>}</span> </span><br/>

<a NAME="E2:237_1"/><b>then </b><i><font color="Green" title="E99">B1</font></i>: 
( <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_1.html#T3" target="_self" title="XXREAL_1:th.3">Th3</a></i>;<br/>
<a NAME="E3:237_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="#E1:237"><i><font color="Green" title="E98">F</font></i></a>, <a class="ref" href="xxreal_0.html#T46" title="XXREAL_0:th.46">XXREAL_0:46</a></i>;<br/>
<b>hence </b><a NAME="E4:237_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E2:237_1"><i><font color="Green" title="E99">B1</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>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> </span>}</span>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">s</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )</span><br/>

<b>assume </b><a NAME="E3:237"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">s</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c3">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:237"/><i><font color="Green" title="E99">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font> )
 ;<br/>
<a NAME="E6:237"/>
<font color="Maroon" title="c3">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/>
<b>hence </b><a NAME="E7:237"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">s</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="#E5:237"><i><font color="Green" title="E99">A2</font></i></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" title="XXREAL_1:th.3">Th3</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>
