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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c1">p</font>,<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E1:213"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c1">p</font>,<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="213_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:213_1"/><span class="lab"><font color="Green" title="E93">A1</font></span>: 
<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#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c1">p</font>,<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">s</font> = <font color="Maroon" title="c2">x</font> as   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:213_1"><span class="lab"><font color="Green" title="E93">A1</font></span></a></span>;<br/>
<a NAME="E4:213_1"/><span class="lab"><font color="Green" title="E94">A2</font></span>: 
<font color="Maroon" title="c1">p</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="#E2:213_1"><span class="lab"><font color="Green" title="E93">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T2" target="_self" onmouseover="rs('xxreal_1/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E5:213_1"/>
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:213_1"><span class="lab"><font color="Green" title="E93">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T2" target="_self" onmouseover="rs('xxreal_1/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E6:213_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:213_1"><span class="lab"><font color="Green" title="E94">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:213_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T5" onmouseover="rs('xxreal_0/T5')" onmouseout="rh()">XXREAL_0:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:213"/>
<span class="p1"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c1">p</font>,<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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