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<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] or ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] &amp; not <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>];<br/>
<a NAME="E2:22_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:22_1_1"/><i><font color="Green">E12</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) ) implies  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] )
 
<b>by </b><i><a class="txt" href="nat_1.html#E1:22_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:22_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> holds <br/> not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>hence </b><a NAME="E3:22_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="nat_1.html#E1:22_1_1"><i><font color="Green">E12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:22_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" target="_self">NAT_1:sch 4</a>(<a class="txt" href="nat_1.html#E2:22_1"><i><font color="Green">E11</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:22_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_1.html#E1:22"><i><font color="Green">E9</font></i></a></i>;<br/>


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