<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">M</font> =  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font>,   <a href="subset_1.html#M1">Element</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> =  <a href="modelc_1.html#K42" target="_self">And_0</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">S</font>;<br/>
<a NAME="E1:109"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="binop_1.html#NM2">BinOp</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font>  st ( for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> holds  <font color="Olive">o1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>,<font color="Olive">g</font>) ) &amp; ( for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> holds  <font color="Olive">o2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>,<font color="Olive">g</font>) ) holds <br/><font color="Olive">o1</font> <a href="funct_2.html#R2">=</a> <font color="Olive">o2</font>
 
<b>from </b><i><a class="ref" href="binop_2.html#S2">BINOP_2:sch 2</a>();<br/></i>
<a NAME="E2:109"/>
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="binop_1.html#NM2">BinOp</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font>  st ( for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> holds <br/><font color="Olive">o1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K42" target="_self">And_0</a> <font color="Olive">f</font>,<font color="Olive">g</font>,<font color="Maroon">S</font> ) &amp; ( for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> holds <br/><font color="Olive">o2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K42" target="_self">And_0</a> <font color="Olive">f</font>,<font color="Olive">g</font>,<font color="Maroon">S</font> ) holds <br/><font color="Olive">o1</font> <a href="funct_2.html#R2">=</a> <font color="Olive">o2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font>, <font color="Maroon">o2</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:109_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">o1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K42" target="_self">And_0</a> <font color="Olive">f</font>,<font color="Olive">g</font>,<font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E2:109_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">o2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K42" target="_self">And_0</a> <font color="Olive">f</font>,<font color="Olive">g</font>,<font color="Maroon">S</font>
 ;<br/>

<a NAME="E3:109_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> holds  <font color="Maroon">o1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>,<font color="Olive">g</font>)
 
<b>by </b><i><a class="txt" href="modelc_1.html#E26"><i><font color="Green">Lemma89</font></i></a></i>;<br/>
<a NAME="E4:109_1"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> holds  <font color="Maroon">o2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>,<font color="Olive">g</font>)
 
<b>by </b><i><a class="txt" href="modelc_1.html#E27"><i><font color="Green">Lemma90</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:109_1"/>
<font color="Maroon">o1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E28"><i><font color="Green">Lemma91</font></i></a>, <a class="txt" href="modelc_1.html#E25"><i><font color="Green">Lemma88</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:109"/>
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="binop_1.html#NM2">BinOp</a> of  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font>  st ( for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> holds <br/><font color="Olive">o1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K42" target="_self">And_0</a> <font color="Olive">f</font>,<font color="Olive">g</font>,<font color="Maroon">S</font> ) &amp; ( for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font> holds <br/><font color="Olive">o2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K42" target="_self">And_0</a> <font color="Olive">f</font>,<font color="Olive">g</font>,<font color="Maroon">S</font> ) holds <br/><font color="Olive">o1</font> <a href="funct_2.html#R2">=</a> <font color="Olive">o2</font>
 ;<br/>


</div>
