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<span class="kw">let </span><font color="Maroon" title="c1">r</font>, <font color="Maroon" title="c2">s</font>, <font color="Maroon" title="c3">p</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">s</font> implies <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">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/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:74"/><span class="lab"><font color="Green" title="E59">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">s</font>
 <span class="kw">and </span><br/><a NAME="E2:74"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></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"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:74"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
not <font color="Maroon" title="c2">s</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">r</font>,<font color="Maroon" title="c2">s</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>;<br/>
<a NAME="E4:74"/>
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="xxreal_1.html#T52" target="_self" title="XXREAL_1:th.52">Th52</a></span>;<br/>
<a NAME="E5:74"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A4</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:74"/>
<font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="xxreal_1.html#T52" target="_self" title="XXREAL_1:th.52">Th52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:74"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E3:74"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E5:74"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="xxreal_1.html#T1" target="_self" title="XXREAL_1:th.1">Th1</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/>


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