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<span class="kw">let </span><font color="Maroon" title="c1">r</font>, <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">q</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font> implies  not <span class="p1"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></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="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:102"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">and </span><br/><a NAME="E2:102"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <span class="p1"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></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="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:102_1"/>
( <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> or <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1"><span class="kw">suppose </span></a><a NAME="E1:102_1_1"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <span class="p1"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></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="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">t</font> being   <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> <span class="kw"> such that </span><br/><a NAME="E3:102_1_1"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">t</font>
 <span class="kw">and </span><br/><a NAME="E4:102_1_1"/><span class="lab"><font color="Green" title="E64">A5</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:102"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="xreal_1.html#T229" title="XREAL_1:th.229">XREAL_1:229</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">t</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="membered.html#D8" title="MEMBERED:def.8">MEMBERED:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span> &amp; not <font color="Maroon" title="c5">t</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="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )</span><br/><span class="kw">thus </span><a NAME="E5:102_1_1"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E4:102_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="xxreal_1.html#T4" target="_self" title="XXREAL_1:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c5">t</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="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/><a NAME="E6:102_1_1"/>
<font color="Maroon" title="c5">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E4:102_1_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:102_1_1"/>
not <font color="Maroon" title="c5">t</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="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T3" target="_self" title="XXREAL_1:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_2"><span class="kw">suppose </span></a><a NAME="E1:102_1_2"/><span class="lab"><font color="Green" title="E62">A6</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <span class="p1"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></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="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c5">t</font> being   <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> <span class="kw"> such that </span><br/><a NAME="E3:102_1_2"/><span class="lab"><font color="Green" title="E63">A7</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">t</font>
 <span class="kw">and </span><br/><a NAME="E4:102_1_2"/><span class="lab"><font color="Green" title="E64">A8</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="ref" href="xreal_1.html#T229" title="XREAL_1:th.229">XREAL_1:229</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">t</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="membered.html#D8" title="MEMBERED:def.8">MEMBERED:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span> &amp; not <font color="Maroon" title="c5">t</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="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )</span><br/><a NAME="E5:102_1_2"/>
<font color="Maroon" title="c5">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102_1_2"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="txt" href="#E4:102_1_2"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:102_1_2"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102_1_2"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="ref" href="xxreal_1.html#T4" target="_self" title="XXREAL_1:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c5">t</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="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/><span class="kw">thus </span><a NAME="E7:102_1_2"/>
not <font color="Maroon" title="c5">t</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="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:102_1_2"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" title="XXREAL_1:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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