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<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">fs</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ;<br/><b>let </b><font color="Maroon" title="c3">E</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:20"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">fs</font>,<font color="Maroon" title="c3">E</font>
 ;<br/>

<a NAME="E2:20"/><b>then </b><i><font color="Green" title="E24">A1</font></i>: 
 ex <font color="Olive" title="b1">e</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">fs</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">E</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c1">p</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">fs</font>,<font color="Maroon" title="c3">E</font> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">ElofE</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">E</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">E</font> = <font color="Maroon" title="c5">ElofE</font>;<br/>
<div><i><font color="Green" title="E25">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1_1"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 ;<br/><a NAME="E2:20_1_1"/><b>then </b>
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:20_1_1"/>
<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:20_1"/>
( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">q</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">q</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">q</font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">q</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> ) )
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c3">E</font><b> such that </b><br/><a NAME="E7:20"/><i><font color="Green" title="E26">A3</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>] implies <font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">q</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>] implies <font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">q</font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5" title="FUNCT_2:sch.5">FUNCT_2:sch 5</a>(<a class="txt" href="zf_fund1.html#E5:20"><i><font color="Green" title="E25">A2</font></i></a>);<br/></i>
<a NAME="E8:20"/>
<a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> , <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4"><i><font color="Green" title="E4">Lm1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">E</font> <b>by </b><i><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">f</font>
;<br/>

<a NAME="E10:20"/><i><font color="Green" title="E27">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E28">A5</font></i>: <font color="Maroon" title="c6">h</font> = 
<font color="Maroon" title="c6">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E4"><i><font color="Green" title="E4">Lm1</font></i></a>, <a class="ref" href="relat_1.html#T77" title="RELAT_1:th.77">RELAT_1:77</a></i>
<br/>.= 
<font color="Maroon" title="c6">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a>  <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E4"><i><font color="Green" title="E4">Lm1</font></i></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> 
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
;<br/>
<a NAME="E12:20"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">fs</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<a NAME="E1:20_3"/><i><font color="Green" title="E29">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">fs</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E2:20"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E10:20"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E2:20_3"/>
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/><font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E2:20"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E7:20"><i><font color="Green" title="E26">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:20_3"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">fs</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:20_3"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:20"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="zf_fund1.html#K3" title="ZF_FUND1:func.3">decode</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">fs</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E11:20"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/>


</div>
