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<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">B</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1"/>
( <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1"/>
<font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>hence </b><a NAME="E3:17_1_1_1_1"/>
<font color="Maroon" title="c2">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_2"/>
not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c3">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:17_1_1_1_2"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">p</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:17_1_1_1_2"/><i><font color="Green" title="E5">A2</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">B</font>
 <b>and </b><br/><a NAME="E6:17_1_1_1_2"/><i><font color="Green" title="E6">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="ref" href="finset_1.html#D1" target="_self" title="FINSET_1:def.1">Def1</a></i>;<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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">B</font>;<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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">x1</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><font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>;<br/><b>consider </b><font color="Maroon" title="c5">q</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:17_1_1_1_2"/><i><font color="Green" title="E7">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">p</font>
 <b>and </b><br/><a NAME="E9:17_1_1_1_2"/><i><font color="Green" title="E8">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">p</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S1" title="PARTFUN1:sch.1">PARTFUN1:sch 1</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:17_1_1_1_2_1"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_1_1_2_1_1"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:17_1_1_1_2_1_1"/><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">p</font>
 <b>and </b><br/><a NAME="E4:17_1_1_1_2_1_1"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E5:17_1_1_1_2"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">x</font>
;<br/>

<b>thus </b><a NAME="E5:17_1_1_1_2_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E8:17_1_1_1_2"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="finset_1.html#E3:17_1_1_1_2_1_1"><i><font color="Green" title="E10">A7</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:17_1_1_1_2_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E9:17_1_1_1_2"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="finset_1.html#E1:17_1_1_1_2_1_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="finset_1.html#E3:17_1_1_1_2_1_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="finset_1.html#E4:17_1_1_1_2_1_1"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:17_1_1_1_2_1"/><i><font color="Green" title="E9">A9</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">q</font>
 <b>and </b><br/><a NAME="E4:17_1_1_1_2_1"/><i><font color="Green" title="E10">A10</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_2_1_2"/>
( <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or  not <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_2_1_2_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E8:17_1_1_1_2"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="finset_1.html#E9:17_1_1_1_2"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="finset_1.html#E3:17_1_1_1_2_1"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="finset_1.html#E4:17_1_1_1_2_1"><i><font color="Green" title="E10">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_2_1_2_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_2_1_2_2"/>
not <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_1_2_1_2_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E3:17_1_1_1_2"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="finset_1.html#E8:17_1_1_1_2"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="finset_1.html#E9:17_1_1_1_2"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="finset_1.html#E3:17_1_1_1_2_1"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="finset_1.html#E4:17_1_1_1_2_1"><i><font color="Green" title="E10">A10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E11:17_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</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="E12:17_1_1_1_2"/>
<font color="Maroon" title="c2">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="finset_1.html#E6:17_1_1_1_2"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="finset_1.html#E8:17_1_1_1_2"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="finset_1.html#D1" target="_self" title="FINSET_1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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