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<span class="kw">let </span><font color="Maroon" title="c1">A</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">PA1</font>, <font color="Olive" title="b2">PA2</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">PA1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b2">PA2</font> holds <br/> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">PA2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">PA1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">PA1</font>, <font color="Maroon" title="c3">PA2</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">PA1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">PA2</font> implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">PA2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">PA1</font> )</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $2;<br/>
<span class="kw">assume </span><a NAME="E1:13"/>
<font color="Maroon" title="c2">PA1</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">PA2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">PA2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">PA1</font></span><br/>

<a NAME="E2:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A1</font></span>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA1</font> holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PA2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">PA1</font>,<font color="Maroon" title="c3">PA2</font><span class="kw"> such that </span><br/><a NAME="E4:13"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E7">A1</font></span></a>);<br/></span>
<span class="kw">assume </span><a NAME="E5:13"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">PA1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">PA2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:13"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">PA1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">PA2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T42" title="NAT_1:th.42">NAT_1:42</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">p2i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:13"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c5">p2i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PA2</font>
 <span class="kw">and </span><br/><a NAME="E9:13"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
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">PA1</font> holds <br/><font color="Maroon" title="c4">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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p2i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T81" title="FINSEQ_4:th.81">FINSEQ_4:81</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">p2i</font> = <font color="Maroon" title="c5">p2i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">PA2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E12:13"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">p2i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T11" target="_self" title="FSM_1:th.11">Th11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">p2q</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c2">PA1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">PA2</font> ;<br/>
<a NAME="E14:13"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
( <font color="Maroon" title="c8">p2q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2i</font> or <font color="Maroon" title="c8">p2q</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">p2i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/>
<a NAME="E15:13"/>
( <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c2">PA1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c2">PA1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">p2q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="borsuk_1.html#D1" title="BORSUK_1:def.1">BORSUK_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:13"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:13"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E12:13"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E14:13"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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