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<b>assume </b><a NAME="E1:3_1_1"/><i><font color="Green" title="E2">A1</font></i>: 
 <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c1">p</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1_1"/>
(  <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c1">p</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st  <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )
 </span><b>by </b><i><a class="txt" href="jordan1g.html#E1:3_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="jordan1g.html#T1" target="_self" title="JORDAN1G:th.1">Th1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1_1"/>
 <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c1">p</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:3_1_1_1_1"/>
contradiction
 ; <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="3_1_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st  <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:3_1_1_1_2"/><i><font color="Green" title="E3">A2</font></i>: 
 <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 ;<br/><font color="Maroon" title="c1">p</font> = 
 <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="jordan1g.html#E3:3_1_1_1_2"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="finseq_6.html#T29" title="FINSEQ_6:th.29">FINSEQ_6:29</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_5.html#T63" title="FINSEQ_5:th.63">FINSEQ_5:63</a></i>
;<br/><b>hence </b><a NAME="E5:3_1_1_1_2"/>
contradiction
 ; <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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