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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>];<br/>
<a NAME="E1:16_1"/>
<font color="Maroon">P<sub>1</sub></font>[0,<font color="Maroon">F<sub>1</sub></font>()]
 
<b>by </b><i><a class="txt" href="nat_1.html#E1:16"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E2:16_1"/><b>then </b><i><font color="Green">E7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:16_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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/>

<b>given </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:16_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

<b>take </b>
<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>)
;<br/>

<b>thus </b><a NAME="E2:16_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1,<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>)]
 <b>by </b><i><a class="txt" href="nat_1.html#E1:16"><i><font color="Green">E6</font></i></a>, <a class="txt" href="nat_1.html#E1:16_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:16_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#S1" target="_self">NAT_1:sch 1</a>(<a class="txt" href="nat_1.html#E2:16_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="nat_1.html#E3:16_1"><i><font color="Green">E8</font></i></a>);</i><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>;<br/>
<a NAME="E5:16_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:16_1_2"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[0,<font color="Maroon">c<sub>1</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[0,<font color="Maroon">c<sub>2</sub></font>] )
 ;<br/>

<a NAME="E2:16_1_2"/>
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not 0 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1 or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] or  not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(0,<font color="Olive">b<sub>2</sub></font>) ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not 0 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1 or  not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] or  not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(0,<font color="Olive">b<sub>2</sub></font>) ) ) )
 
;<br/>
<a NAME="E3:16_1_2"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="txt" href="nat_1.html#E1:16"><i><font color="Green">E6</font></i></a>, <a class="txt" href="nat_1.html#E1:16_1_2"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:16_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:16_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> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><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/>

<b>assume </b><a NAME="E1:16_1_3"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E2:16_1_3"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1,<font color="Maroon">c<sub>2</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1,<font color="Maroon">c<sub>3</sub></font>] )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:16_1_3"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>] &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>,<font color="Maroon">c<sub>5</sub></font>) )
 <b>by </b><i><a class="txt" href="nat_1.html#E1:16"><i><font color="Green">E6</font></i></a>, <a class="txt" href="nat_1.html#E2:16_1_3"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E5:16_1_3"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1" target="_self">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font>) )
 
<b>by </b><i><a class="txt" href="nat_1.html#E1:16"><i><font color="Green">E6</font></i></a>, <a class="txt" href="nat_1.html#E2:16_1_3"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:16_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:16_1_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="nat_1.html#E4:16_1_3"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E7:16_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>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" target="_self">NAT_1:sch 1</a>(<a class="txt" href="nat_1.html#E5:16_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="nat_1.html#E6:16_1"><i><font color="Green">E10</font></i></a>);</i><br/>


</div>
