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<span class="kw">let </span><font color="Maroon" title="c1">x</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">x</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>  implies ( <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">x</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="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c1">x</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">x</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="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:33_1"/>
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ; <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="E2:33_1"/><span class="kw">then </span>
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D3" target="_self" title="XXREAL_3:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:33_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E25">A1</font></span></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>
<span class="kw">hence </span><a NAME="E3:33"/>
<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>; <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">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><br/>

<span class="kw">thus </span><a NAME="E4:33"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a>, <a class="ref" href="xxreal_3.html#T5" target="_self" title="XXREAL_3:th.5">Th5</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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