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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>assume </b><a NAME="E1:9"/><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>
 ;<br/>

<b>assume </b><a NAME="E2:9"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">PA1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">PA2</font>
 ;<br/>

<a NAME="E3:9"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">PA1</font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">PA2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T73">CARD_1:73</a></i>;<br/>
<a NAME="E4:9"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">PA1</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">PA2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#D11">CARD_1:def 11</a></i>;<br/>
<a NAME="E5:9"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">PA1</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">PA2</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D11">CARD_1:def 11</a></i>;<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/>
<a NAME="E6:9"/><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/> 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="E8:9"/><i><font color="Green">E39</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>
<b>consider </b><font color="Maroon">p2i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:9"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p2i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA2</font>
 <b>and </b><br/><a NAME="E11:9"/><i><font color="Green">E42</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">PA1</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2i</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T81">FINSEQ_4:81</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p2i</font> = <font color="Maroon">p2i</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA2</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E14:9"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p2i</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p2q</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K2">proj</a> <font color="Maroon">PA1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA2</font> ;<br/>
<a NAME="E16:9"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K2">proj</a> <font color="Maroon">PA1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D1">BORSUK_1:def 1</a></i>;<br/>
<a NAME="E17:9"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K2">proj</a> <font color="Maroon">PA1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p2q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:9"/>
( <font color="Maroon">p2q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2i</font> or <font color="Maroon">p2q</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">p2i</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="E19:9"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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