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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">m</font> being   <a href="pnproc_1.html#M1" target="_self">marking</a> of <font color="Maroon">P</font> st <br/>( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K8" target="_self">fire</a> <font color="Maroon">t</font>,<font color="Olive">m</font> );<br/>
<a NAME="E1:54_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">P</font>,<a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E2:54_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">P</font>,<a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">P</font>,<a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as    <a href="pnproc_1.html#M1" target="_self">marking</a> of <font color="Maroon">P</font> <b>by </b><i/>;<br/>
<b>take </b>
 <a href="pnproc_1.html#K8" target="_self">fire</a> <font color="Maroon">t</font>,<font color="Maroon">m</font>
;<br/>

<b>thus </b><a NAME="E3:54_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>, <a href="pnproc_1.html#K8" target="_self">fire</a> <font color="Maroon">t</font>,<font color="Maroon">m</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:54_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">P</font>,<a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E5:54_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">P</font>,<a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="pnproc_1.html#E5:54_1_1"><i><font color="Green">E37</font></i></a>, );<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E6:54_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">P</font>,<a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="pnproc_1.html#M1" target="_self">marking</a> of <font color="Maroon">P</font>;<br/>

<a NAME="E7:54_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">P</font>,<a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E8:54_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>]
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E9:54_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K8" target="_self">fire</a> <font color="Maroon">t</font>,<font color="Maroon">m</font>
 ;<br/>


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