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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">r</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font> implies <span class="p1">(<span class="default"><span class="p2"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">r</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:180"/><i><font color="Green" title="E77">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font>
 <b>and </b><br/><a NAME="E2:180"/><i><font color="Green" title="E78">A2</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font>
 <b>and </b><br/><a NAME="E3:180"/><i><font color="Green" title="E79">A3</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font>
 <b>and </b><br/><a NAME="E4:180"/><i><font color="Green" title="E80">A4</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font>
 ; <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"><span class="p2"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">r</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:180_1"/>
( <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font> or <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_1_1"><b>suppose </b></a><a NAME="E1:180_1_1"/>
<font color="Maroon" title="c2">r</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">r</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">r</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> = 
<span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
<b>by </b><i><a class="txt" href="#E1:180"><i><font color="Green" title="E77">A1</font></i></a>, <a class="ref" href="xxreal_1.html#T166" target="_self" title="XXREAL_1:th.166">Th166</a></i>
<br/>.= 
<span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="#E2:180"><i><font color="Green" title="E78">A2</font></i></a>, <a class="txt" href="#E4:180"><i><font color="Green" title="E80">A4</font></i></a>, <a class="ref" href="xxreal_1.html#T167" target="_self" title="XXREAL_1:th.167">Th167</a></i>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="180_1_2"><b>suppose </b></a><a NAME="E1:180_1_2"/><i><font color="Green" title="E81">A5</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 ; <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"><span class="p2"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">r</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">r</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">r</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
<b>by </b><i><a class="ref" href="xxreal_1.html#T29" target="_self" title="XXREAL_1:th.29">Th29</a></i>
<br/>.= 
<span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="#E2:180"><i><font color="Green" title="E78">A2</font></i></a>, <a class="txt" href="#E3:180"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="#E1:180_1_2"><i><font color="Green" title="E81">A5</font></i></a>, <a class="ref" href="xxreal_1.html#T175" target="_self" title="XXREAL_1:th.175">Th175</a></i>
;<br/> <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></div><b>end;</b></div>

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