<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">PX</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:8"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">PX</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:8"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">PX</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T73" title="CARD_1:th.73">CARD_1:73</a></i>;<br/>
<a NAME="E3:8"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">PX</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#D11" title="CARD_1:def.11">CARD_1:def 11</a></i>;<br/>
<a NAME="E4:8"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">PX</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D11" title="CARD_1:def.11">CARD_1:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">Pi</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:8"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c3">Pi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PX</font>
 <b>and </b><br/><a NAME="E7:8"/><i><font color="Green" title="E8">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="c1">X</font> holds <br/><span class="p1">(<span class="default"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">proj</a> <font color="Maroon" title="c2">PX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Pi</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T81" title="FINSEQ_4:th.81">FINSEQ_4:81</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">Pi</font> = <font color="Maroon" title="c3">Pi</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PX</font> <b>by </b><i><a class="txt" href="fsm_1.html#E6:8"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E10:8"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Pi</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#T11" target="_self" title="FSM_1:th.11">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Pq</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">proj</a> <font color="Maroon" title="c2">PX</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">q</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PX</font> ;<br/>
<a NAME="E12:8"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Pq</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D1" title="BORSUK_1:def.1">BORSUK_1:def 1</a></i>;<br/>
<a NAME="E13:8"/>
( <font color="Maroon" title="c6">Pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Pi</font> or <font color="Maroon" title="c6">Pq</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Pi</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E14:8"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_1.html#E7:8"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="fsm_1.html#E10:8"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="fsm_1.html#E12:8"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div>
