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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> implies  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:7"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> iff <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:7_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font></span><br/><b>then consider </b><font color="Maroon" title="c4">X'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E3:7_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c4">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c5">X'</font> = <font color="Maroon" title="c4">X'</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">X'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">X'</font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>
<br/>.= 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
<b>by </b><i><a class="txt" href="ramsey_1.html#E3:7_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="ramsey_1.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a></i>
;<br/><a NAME="E6:7_1_1"/><b>then </b>
<font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/><a NAME="E7:7_1_1"/><b>then </b>
<font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">X'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E8:7_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:7_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:7_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><br/>

<a NAME="E3:7_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c3">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<a NAME="E5:7_1"/>
<font color="Maroon" title="c4">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="ramsey_1.html#E2:7_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:7_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:7"/>
 <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">Y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
