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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> ex <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1,$1] );<br/>
<a NAME="E1:7_1"/><i><font color="Green" title="E7">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>given </b><font color="Maroon" title="c2">S</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <b> such that </b><a NAME="E2:7_1_1"/><i><font color="Green" title="E8">A5</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">S</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>] )
 ;<br/>

<b>take </b><font color="Maroon" title="c3">S'</font> = <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c1">n</font>);<br/>

<a NAME="E3:7_1_1"/>
( <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c1">n</font>) )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E2:7"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="circcmb2.html#E2:7_1_1"><i><font color="Green" title="E8">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_1"/>
( <font color="Maroon" title="c3">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">S'</font>,<font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )
 <b>by </b><i><a class="txt" href="circcmb2.html#E3:7"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="circcmb2.html#E2:7_1_1"><i><font color="Green" title="E8">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:7_1"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="circcmb2.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:7_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="circcmb2.html#E2:7_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="circcmb2.html#E1:7_1"><i><font color="Green" title="E7">A4</font></i></a>);</i><br/>


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