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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">o</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  ex <font color="Olive">r</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Olive">l</font>,<font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D5">MONOID_0:def 5</a><br/>

<b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D</font>, <font color="Maroon">g</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D5">MONOID_0:def 5</a><br/>

<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><font color="Maroon">o</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:23"/><i><font color="Green">E44</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> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_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="E1:23_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>, <font color="Maroon">l</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E4:23_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">l</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>thus </b><a NAME="E5:23_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">r</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">o</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:23"/><i><font color="Green">E46</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> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><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="E1:23_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>, <font color="Maroon">l</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E4:23_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">l</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">l</font>
;<br/>

<b>thus </b><a NAME="E5:23_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">l</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">h1</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">D</font><b> such that </b><br/><a NAME="E5:23"/><i><font color="Green">E48</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> <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="monoid_1.html#S1" target="_self">MONOID_1:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">h2</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">D</font><b> such that </b><br/><a NAME="E7:23"/><i><font color="Green">E80</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> <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="monoid_1.html#S1" target="_self">MONOID_1:sch 1</a>();<br/></i>
<a NAME="E8:23"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h1</font>, <font color="Maroon">h2</font> = <font color="Maroon">h2</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">D</font> <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">h1</font>
;<br/>

<b>take </b>
<font color="Maroon">h2</font>
;<br/>

<a NAME="E10:23"/><i><font color="Green">E81</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">h2</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">f</font>,<font color="Maroon">h1</font> &amp; <font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">h2</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">o</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">h2</font>,<font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><b>now </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="E1:23_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">f</font>,<font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">o</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:23"/>
<span class="p1">(<span class="default"><font color="Maroon">o</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">f</font>,<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><b>now </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="E1:23_4"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K3" target="_self">.:</a> <font color="Maroon">h2</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">o</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:23"/>
<span class="p1">(<span class="default"><font color="Maroon">o</font>,<font color="Maroon">D</font> <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">h2</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
