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<b>let </b><font color="Maroon" title="c1">m</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b2">X</font> being <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/> for <font color="Olive" title="b3">P</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Olive" title="b2">X</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/> ex <font color="Olive" title="b4">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b2">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b4">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b4">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Olive" title="b3">P</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1"/>
( <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 or <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/>
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b2">X</font> being <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/> for <font color="Olive" title="b3">P</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Olive" title="b2">X</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/> ex <font color="Olive" title="b4">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b2">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b4">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b4">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Olive" title="b3">P</font> )</span><br/><div class="add"><a NAME="E2:24_1_1"/><b>then </b>
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 ;<br/><a NAME="E3:24_1_1"/><b>then </b><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c2">r</font> = 1;<br/><b>take </b>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being <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/> for <font color="Olive" title="b2">P</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Olive" title="b1">X</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/> ex <font color="Olive" title="b3">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b3">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Olive" title="b2">P</font> )</span><br/><b>let </b><font color="Maroon" title="c3">X</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"> for <font color="Olive" title="b1">P</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c3">X</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/> ex <font color="Olive" title="b2">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b2">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Olive" title="b1">P</font> )</span><br/><b>let </b><font color="Maroon" title="c4">P</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c3">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies  ex <font color="Olive" title="b1">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b1">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Maroon" title="c4">P</font> ) )</span><br/><b>assume </b><a NAME="E4:24_1_1"/><i><font color="Green" title="E20">A2</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or  ex <font color="Olive" title="b1">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b1">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Maroon" title="c4">P</font> ) )</span><br/><b>assume </b><a NAME="E5:24_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b1">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Maroon" title="c4">P</font> )</span><br/><b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:24_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E4:24_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">S</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E7:24_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>take </b>
<font color="Maroon" title="c6">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c6">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Maroon" title="c4">P</font> )</span><br/><a NAME="E9:24_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>;<br/><b>thus </b><a NAME="E10:24_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:24_1_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Maroon" title="c4">P</font></span><br/><a NAME="E11:24_1_1"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">P</font> st  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c7">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">P</font>;<br/>
<b>take </b>
<font color="Maroon" title="c7">p</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> 2,<font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">p</font></span><br/>

<a NAME="E2:24_1_1_1"/>
 <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c6">S</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="txt" href="ramsey_1.html#E9:24_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="ramsey_1.html#T6" target="_self" title="RAMSEY_1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1_1_1"/>
 <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</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>hence </b><a NAME="E12:24_1_1"/>
<font color="Maroon" title="c6">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="ramsey_1.html#D1" target="_self" title="RAMSEY_1:def.1">Def1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><b>suppose </b></a><a NAME="E1:24_1_2"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b2">X</font> being <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/> for <font color="Olive" title="b3">P</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Olive" title="b2">X</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/> ex <font color="Olive" title="b4">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b2">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b4">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b4">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Olive" title="b3">P</font> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c2">r</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:24_1_2"/><i><font color="Green" title="E20">A6</font></i>: 
( <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 <b>and </b><br/><a NAME="E4:24_1_2"/><i><font color="Green" title="E21">A7</font></i>: 
for <font color="Olive" title="b1">X</font> being <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/> for <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Olive" title="b1">X</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font> holds <br/> ex <font color="Olive" title="b3">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">X</font> st <br/>( (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Olive" title="b3">S</font></span>)</span></span>)</span> <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">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Olive" title="b3">S</font></span>)</span></span>)</span> <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">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 <b>by </b><i><a class="ref" href="ramsey_1.html#T16" target="_self" title="RAMSEY_1:th.16">Th16</a></i>;<br/><b>take </b>
<font color="Maroon" title="c2">r</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being <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/> for <font color="Olive" title="b2">P</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Olive" title="b1">X</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/> ex <font color="Olive" title="b3">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b3">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Olive" title="b2">P</font> )</span><br/><b>let </b><font color="Maroon" title="c3">X</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"> for <font color="Olive" title="b1">P</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c3">X</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/> ex <font color="Olive" title="b2">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b2">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Olive" title="b1">P</font> )</span><br/><b>let </b><font color="Maroon" title="c4">P</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c3">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies  ex <font color="Olive" title="b1">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b1">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Maroon" title="c4">P</font> ) )</span><br/><b>assume </b><a NAME="E5:24_1_2"/><i><font color="Green" title="E22">A8</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">X</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">r</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X</font> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b1">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Maroon" title="c4">P</font> )</span><br/><a NAME="E6:24_1_2"/><b>then </b>
2 <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="c3">X</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:24_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:24_1_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span> <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="c3">X</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>;<br/><a NAME="E8:24_1_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></i>;<br/><a NAME="E9:24_1_2"/><b>then </b><i><font color="Green" title="E23">A9</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T76" title="FINSEQ_1:th.76">FINSEQ_1:76</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">X'</font> =  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c3">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="ramsey_1.html#E5:24_1_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="ramsey_1.html#E3:24_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="txt" href="ramsey_1.html#E9:24_1_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="group_10.html#T2" title="GROUP_10:th.2">GROUP_10:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">P'</font> = <font color="Maroon" title="c4">P</font> as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c5">X'</font> ;<br/><a NAME="E12:24_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E5:24_1_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>;<br/><a NAME="E13:24_1_2"/><b>then </b><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c6">P'</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">F1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E15:24_1_2"/><i><font color="Green" title="E25">A11</font></i>: 
( <font color="Maroon" title="c7">F1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P'</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 )
 <b>by </b><i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">F1</font> = <font color="Maroon" title="c7">F1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">P'</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 <b>by </b><i><a class="txt" href="ramsey_1.html#E15:24_1_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/><b>set </b><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c8">F1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c6">P'</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c10">F</font> = <font color="Maroon" title="c8">F1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c6">P'</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c3">X</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 ;<br/><b>consider </b><font color="Maroon" title="c11">S</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">X</font><b> such that </b><br/><a NAME="E19:24_1_2"/><i><font color="Green" title="E26">A12</font></i>: 
( (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font></span>)</span></span>)</span> <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">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font></span>)</span></span>)</span> <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">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E4:24_1_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="ramsey_1.html#E5:24_1_2"><i><font color="Green" title="E22">A8</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c11">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Maroon" title="c4">P</font> )</span><br/><b>thus </b><a NAME="E20:24_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">S</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E19:24_1_2"><i><font color="Green" title="E26">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Maroon" title="c4">P</font></span><br/><b>consider </b><font color="Maroon" title="c12">h</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font>;<br/><a NAME="E22:24_1_2"/><i><font color="Green" title="E27">A13</font></i>: 
 <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3"><i><font color="Green" title="E3">Lm1</font></i></a></i>;<br/><a NAME="E23:24_1_2"/>
2 <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="c11">S</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:24_1_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="ramsey_1.html#E19:24_1_2"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E24:24_1_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span> <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="c11">S</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>;<br/><a NAME="E25:24_1_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">S</font>
 <b>by </b><i><a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></i>;<br/><a NAME="E26:24_1_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">S</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T76" title="FINSEQ_1:th.76">FINSEQ_1:76</a></i>;<br/><a NAME="E27:24_1_2"/><b>then </b><i><font color="Green" title="E28">A14</font></i>: 
not  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:24_1_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="ramsey_1.html#E19:24_1_2"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="group_10.html#T2" title="GROUP_10:th.2">GROUP_10:2</a></i>;<br/><a NAME="E28:24_1_2"/><b>then </b>
<font color="Maroon" title="c12">h</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> 2,<font color="Maroon" title="c11">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c13">h</font> = <font color="Maroon" title="c12">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X'</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E22:24_1_2"><i><font color="Green" title="E27">A13</font></i></a></i>;<br/><a NAME="E30:24_1_2"/><i><font color="Green" title="E29">A15</font></i>: 
<font color="Maroon" title="c10">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_1_2_1"/><i><font color="Green" title="E30">A16</font></i>: 
( 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 &amp; 2 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E2:24_1_2_1"/>
<font color="Maroon" title="c10">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font></span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E22:24_1_2"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1_2_1"/>
<font color="Maroon" title="c10">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:24_1_2_1"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="txt" href="ramsey_1.html#E27:24_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="ramsey_1.html#E19:24_1_2"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="ramsey_1.html#E10"><i><font color="Green" title="E10">Lm4</font></i></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>set </b><font color="Maroon" title="c14">E</font> =  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c6">P'</font>;<br/><b>reconsider </b><font color="Maroon" title="c15">E</font> =  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c6">P'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">P</font> <b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/><a NAME="E32:24_1_2"/>
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>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">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="c16">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> 2,<font color="Maroon" title="c11">S</font> implies <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font> )</span><br/>

<b>assume </b><a NAME="E1:24_1_2_2"/><i><font color="Green" title="E30">A17</font></i>: 
<font color="Maroon" title="c16">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> 2,<font color="Maroon" title="c11">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c17">x'</font> = <font color="Maroon" title="c16">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c3">X</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E22:24_1_2"><i><font color="Green" title="E27">A13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">x''</font> = <font color="Maroon" title="c16">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X'</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E1:24_1_2_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="txt" href="ramsey_1.html#E22:24_1_2"><i><font color="Green" title="E27">A13</font></i></a></i>;<br/>
<a NAME="E4:24_1_2_2"/><i><font color="Green" title="E31">A18</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<a NAME="E5:24_1_2_2"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E13:24_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="card_1.html#T46" title="CARD_1:th.46">CARD_1:46</a></i>;<br/>
<a NAME="E6:24_1_2_2"/><b>then </b><i><font color="Green" title="E32">A19</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:24_1_2_2"/>
<font color="Maroon" title="c17">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E6:24_1_2_2"><i><font color="Green" title="E32">A19</font></i></a>, <a class="txt" href="ramsey_1.html#E1:24_1_2_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:24_1_2_2"/><b>then </b><i><font color="Green" title="E33">A20</font></i>: 
<font color="Maroon" title="c17">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E9:24_1_2_2"/>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E6:24_1_2_2"><i><font color="Green" title="E32">A19</font></i></a>, <a class="txt" href="ramsey_1.html#E27:24_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:24_1_2_2"/><b>then </b><i><font color="Green" title="E34">A21</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<i><font color="Green" title="E35">A22</font></i>: <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x'</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E1:24_1_2_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">h</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E30:24_1_2"><i><font color="Green" title="E29">A15</font></i></a>, <a class="txt" href="ramsey_1.html#E8:24_1_2_2"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="ramsey_1.html#E10:24_1_2_2"><i><font color="Green" title="E34">A21</font></i></a>, <a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>
<br/>.= 
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">h</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E27:24_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<i><font color="Green" title="E36">A23</font></i>: <font color="Maroon" title="c8">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c6">P'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x'</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c6">P'</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">h</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E11:24_1_2_2"><i><font color="Green" title="E35">A22</font></i></a>, <a class="txt" href="ramsey_1.html#E6:24_1_2_2"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c8">F1</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="c6">P'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">h</font></span>)</span>
<b>by </b><i><a class="txt" href="ramsey_1.html#E6:24_1_2_2"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/>
<a NAME="E13:24_1_2_2"/><i><font color="Green" title="E37">A24</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c6">P'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P'</font>
 
;<br/>
<a NAME="E14:24_1_2_2"/>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c6">P'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c6">P'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x'</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E13:24_1_2_2"><i><font color="Green" title="E37">A24</font></i></a>, <a class="txt" href="ramsey_1.html#E12:24_1_2_2"><i><font color="Green" title="E36">A23</font></i></a>, <a class="txt" href="ramsey_1.html#E15:24_1_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E15:24_1_2_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E4:24_1_2_2"><i><font color="Green" title="E31">A18</font></i></a>, <a class="ref" href="ramsey_1.html#T13" target="_self" title="RAMSEY_1:th.13">Th13</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><a NAME="E33:24_1_2"/><b>then </b>
 <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> 2,<font color="Maroon" title="c11">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">E</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E34:24_1_2"/>
<font color="Maroon" title="c11">S</font> <a href="ramsey_1.html#R1" title="RAMSEY_1:pred.1">is_homogeneous_for</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="ramsey_1.html#D1" target="_self" title="RAMSEY_1:def.1">Def1</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></div><b>end;</b></div>

</div>
