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<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">PA1</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">PA2</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">A</font>;<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">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:11"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">PA1</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">PA2</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">PA1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">PA2</font> )
 ;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green">E23</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA1</font> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>] )
 
<b>by </b><i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">PA1</font>,<font color="Maroon">PA2</font><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E5:11"/><i><font color="Green">E39</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PA1</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA2</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E6:11"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">PA2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p2'</font> = <font color="Maroon">p2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA2</font> ;<br/>
<b>consider </b><font color="Maroon">p1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA1</font><b> such that </b><br/><a NAME="E4:11_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p2'</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:11_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:11_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p2'</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:11_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p1</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:11_1_1"/>
<font color="Maroon">p2'</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">fp1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:11_1"/>
<font color="Maroon">p2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fp1</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E9:11_1"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:11"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PA2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E8:11"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T79">FINSEQ_4:79</a></i>;<br/>
<a NAME="E9:11"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA1</font> holds  <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA1</font>;<br/>

<a NAME="E1:11_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:11_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">fx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
;<br/>
<b>assume </b><a NAME="E4:11_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:11_2"/><i><font color="Green">E82</font></i>: 
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>consider </b><font color="Maroon">E1</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E8:11_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">PA1</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E1</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T43">EQREL_1:43</a></i>;<br/>
<a NAME="E9:11_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">E1</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">CE1a</font> =  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">E1</font>,<font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA1</font> <b>by </b><i>, , <a class="ref" href="fsm_1.html#T1" target="_self">Th1</a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E11:11_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">CE1a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">CE1a</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">fCE1a</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">CE1a</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E13:11_2"/>
<font color="Maroon">fx</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">fCE1a</font>
 
<b>by </b><i>, , , <a class="txt" href="fsm_1.html#E1:11"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E14:11_2"/><b>then </b>
<font color="Maroon">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fCE1a</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E15:11_2"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_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/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:11_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA1</font> ;<br/><b>set </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>;<br/><a NAME="E3:11_3_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA2</font>
 ;<br/><b>hence </b><a NAME="E4:11_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA2</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:11_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA2</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:11_3"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <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">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA1</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:11_3"/>
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:11_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:11"/>
<font color="Maroon">PA1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PA2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
