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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>





<b>assume </b><a NAME="E1:99"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:99_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E3:99_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i>, , </i>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">g</font>;<br/><b>thus </b><a NAME="E5:99_1"/>
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="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> = <font color="Maroon">y</font>;<br/><a NAME="E3:99_2"/><i><font color="Green">E34</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">X</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) )
 <b>by </b><i>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:99_2"/><i><font color="Green">E35</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">X</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5" target="_self">FUNCT_2:sch 5</a>(<a class="txt" href="funct_2.html#E3:99_2"><i><font color="Green">E34</font></i></a>);<br/></i><b>take </b><font color="Maroon">g</font> = <font color="Maroon">g</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E6:99_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><b>hence </b><a NAME="E7:99_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="funct_2.html#E5:99_2"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:99"/>
 ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> st <br/>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="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <b>by </b><i/>;<br/>


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