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<b>let </b><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">t</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>assume </b><a NAME="E1:233"/><i><font color="Green" title="E84">A1</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font>
 ;<br/>

<a NAME="E2:233"/><b>then </b><i><font color="Green" title="E85">A2</font></i>: 
( <font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a>, <a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="233_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:233_1"/>
( ( <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) or ( <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  &amp; <font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or ( <font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) or ( <font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) )
 </span><b>by </b><i><a class="txt" href="xreal_1.html#E2:233"><i><font color="Green" title="E85">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="233_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:233_1_1"/><i><font color="Green" title="E86">A3</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 <b>and </b><br/><a NAME="E2:233_1_1"/><i><font color="Green" title="E87">A4</font></i>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c4">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>take </b>
<font color="Maroon" title="c4">s</font>
;<br/><b>thus </b><a NAME="E4:233_1_1"/>
( <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> )
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:233_1_1"><i><font color="Green" title="E86">A3</font></i></a>, <a class="txt" href="xreal_1.html#E2:233_1_1"><i><font color="Green" title="E87">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="233_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:233_1_2"/><i><font color="Green" title="E86">A5</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 <b>and </b><br/><a NAME="E2:233_1_2"/><i><font color="Green" title="E87">A6</font></i>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c4">t</font> = <font color="Maroon" title="c3">t</font> as  <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>by </b><i><a class="txt" href="xreal_1.html#E2:233_1_2"><i><font color="Green" title="E87">A6</font></i></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">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> <b> such that </b><br/><a NAME="E5:233_1_2"/><i><font color="Green" title="E88">A7</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T4" target="_self" title="XREAL_1:th.4">Th4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">s</font>
;<br/><a NAME="E6:233_1_2"/>
<font color="Maroon" title="c5">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>hence </b><a NAME="E7:233_1_2"/>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:233_1_2"><i><font color="Green" title="E86">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/><b>thus </b><a NAME="E8:233_1_2"/>
<font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:233_1_2"><i><font color="Green" title="E88">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="233_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:233_1_3"/><i><font color="Green" title="E86">A8</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>and </b><br/><a NAME="E2:233_1_3"/><i><font color="Green" title="E87">A9</font></i>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c4">r'</font> = <font color="Maroon" title="c2">r</font> as  <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>by </b><i><a class="txt" href="xreal_1.html#E1:233_1_3"><i><font color="Green" title="E86">A8</font></i></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">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> <b> such that </b><br/><a NAME="E5:233_1_3"/><i><font color="Green" title="E88">A10</font></i>: 
<font color="Maroon" title="c4">r'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T3" target="_self" title="XREAL_1:th.3">Th3</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">s</font>
;<br/><b>thus </b><a NAME="E6:233_1_3"/>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:233_1_3"><i><font color="Green" title="E88">A10</font></i></a></i>;<br/><a NAME="E7:233_1_3"/>
<font color="Maroon" title="c5">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>hence </b><a NAME="E8:233_1_3"/>
<font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:233_1_3"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="233_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:233_1_4"/><i><font color="Green" title="E86">A11</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>and </b><br/><a NAME="E2:233_1_4"/><i><font color="Green" title="E87">A12</font></i>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c4">r</font> = <font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c3">t</font> as  <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>by </b><i><a class="txt" href="xreal_1.html#E1:233_1_4"><i><font color="Green" title="E86">A11</font></i></a>, <a class="txt" href="xreal_1.html#E2:233_1_4"><i><font color="Green" title="E87">A12</font></i></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">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> <b> such that </b><br/><a NAME="E5:233_1_4"/><i><font color="Green" title="E88">A13</font></i>: 
( <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">t</font> )
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:233"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="xreal_1.html#T7" target="_self" title="XREAL_1:th.7">Th7</a></i>;<br/><b>take </b>
<font color="Maroon" title="c6">s</font>
;<br/><b>thus </b><a NAME="E6:233_1_4"/>
( <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">t</font> )
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:233_1_4"><i><font color="Green" title="E88">A13</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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