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<b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E2">A2</font></i>: 
( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] or  ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">b</font>] )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1"/>
(  ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">b</font>] or for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] )
 </span><b>by </b><i><a class="txt" href="schems_1.html#E1:10_1"><i><font color="Green" title="E2">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1"/>
not for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">b</font>]
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c1">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:10_1_1_1"/><i><font color="Green" title="E3">A3</font></i>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">b</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1_1_1_1"/>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font>
 ;<br/><a NAME="E2:10_1_1_1_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">b</font>] )
 <b>by </b><i><a class="txt" href="schems_1.html#E1:10"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:10_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="schems_1.html#E3:10_1_1_1"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:10_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_2"/><i><font color="Green" title="E3">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>consider </b><font color="Maroon" title="c2">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:10_1_1_2_1"/><i><font color="Green" title="E4">A5</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">a</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">b</font>] )
 <b>by </b><i><a class="txt" href="schems_1.html#E1:10"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/><b>thus </b><a NAME="E3:10_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="schems_1.html#E1:10_1_1_2"><i><font color="Green" title="E3">A4</font></i></a>, <a class="txt" href="schems_1.html#E2:10_1_1_2_1"><i><font color="Green" title="E4">A5</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:10_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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