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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> = <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:4"/><i><font color="Green" title="E5">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green" title="E6">A2</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> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c3">f</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>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c3">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E4:4"/>
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4_1"/><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c4">x1</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="c3">f</font> &amp; <font color="Maroon" title="c5">x2</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="c3">f</font> )
 <b>and </b><br/><a NAME="E2:4_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">X1</font> = <font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c7">X2</font> = <font color="Maroon" title="c5">x2</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="yellow_8.html#E2:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="yellow_8.html#E1:4_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<font color="Maroon" title="c6">X1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  = 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font>
<b>by </b><i><a class="txt" href="yellow_8.html#E2:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="yellow_8.html#E3:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="yellow_8.html#E1:4_1"><i><font color="Green" title="E7">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">X2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="txt" href="yellow_8.html#E2:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="yellow_8.html#E3:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="yellow_8.html#E1:4_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="yellow_8.html#E2:4_1"><i><font color="Green" title="E8">A4</font></i></a></i>
;<br/>
<b>hence </b><font color="Maroon" title="c4">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">X2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 

<br/>.= 
<font color="Maroon" title="c5">x2</font>

;<br/><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:4"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="yellow_8.html#E2:4"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c2">F</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_2"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:4_2"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c5">u</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="c3">f</font>
 <b>and </b><br/><a NAME="E4:4_2"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u</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>reconsider </b><font color="Maroon" title="c6">Y</font> = <font color="Maroon" title="c5">u</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="yellow_8.html#E2:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="yellow_8.html#E3:4_2"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/>
<a NAME="E6:4_2"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="yellow_8.html#E2:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="yellow_8.html#E3:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="yellow_8.html#E3:4_2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="yellow_8.html#E4:4_2"><i><font color="Green" title="E8">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:4_2"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="yellow_8.html#E2:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="yellow_8.html#E3:4_2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="setfam_1.html#T49" title="SETFAM_1:th.49">SETFAM_1:49</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E7:4"/><i><font color="Green" title="E7">A7</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c4">e</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E9:4"/><i><font color="Green" title="E8">A8</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="yellow_8.html#E7:4"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<font color="Maroon" title="c4">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 

<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="yellow_8.html#E3:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="yellow_8.html#E9:4"><i><font color="Green" title="E8">A8</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E11:4"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="yellow_8.html#E2:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="yellow_8.html#E9:4"><i><font color="Green" title="E8">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div>
