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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</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"> for <font color="Olive" title="b1">X</font> being <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</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> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">X</font>  st  <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> <font color="Maroon" title="c2">k</font> holds <br/> ex <font color="Olive" title="b3">H</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/>( not <font color="Olive" title="b3">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">H</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#NV2" title="FINSET_1:NV.2">infinite</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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">X</font>  st  <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> <font color="Maroon" title="c2">k</font> holds <br/> ex <font color="Olive" title="b2">H</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/>( not <font color="Olive" title="b2">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b2">H</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> <font color="Maroon" title="c1">n</font>,<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#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> <font color="Maroon" title="c2">k</font> implies  ex <font color="Olive" title="b1">H</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/>( not <font color="Olive" title="b1">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">H</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="E1:21"/>
 <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> <font color="Maroon" title="c2">k</font>
 ; <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">H</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/>( not <font color="Olive" title="b1">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">H</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="E2:21"/><b>then </b><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">k</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">F1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:21"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c5">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="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> )
 <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="c6">F1</font> = <font color="Maroon" title="c5">F1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">k</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E4:21"><i><font color="Green" title="E19">A2</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="c7">F</font> = <font color="Maroon" title="c6">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="c4">P</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c6">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="c4">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">k</font> ;<br/>
<a NAME="E7:21"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E2:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="card_1.html#T46" title="CARD_1:th.46">CARD_1:46</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">H</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="E9:21"/><i><font color="Green" title="E21">A4</font></i>: 
( not <font color="Maroon" title="c9">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c8">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</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#E7:21"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="ramsey_1.html#T14" target="_self" title="RAMSEY_1:th.14">Th14</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">H</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c9">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c9">H</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="E10:21"/>
not <font color="Maroon" title="c9">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E9:21"><i><font color="Green" title="E21">A4</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="c9">H</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="c10">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</font>;<br/>
<a NAME="E12:21"/><i><font color="Green" title="E22">A5</font></i>: 
 <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</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> <font color="Maroon" title="c1">n</font>,<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="E13:21"/>
not  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E9:21"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a></i>;<br/>
<a NAME="E14:21"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">n</font> <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="c9">H</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T13" title="CARD_4:th.13">CARD_4:13</a></i>;<br/>
<a NAME="E15:21"/><b>then </b><i><font color="Green" title="E23">A6</font></i>: 
not  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</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#E9:21"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="group_10.html#T2" title="GROUP_10:th.2">GROUP_10:2</a></i>;<br/>
<a NAME="E16:21"/><b>then </b>
<font color="Maroon" title="c10">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c10">h</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">X</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E12:21"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">E</font> =  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c4">P</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">E</font> =  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c4">P</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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="E19:21"/>
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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</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="c13">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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="c14">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">n</font>,<font color="Maroon" title="c9">H</font> implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">E</font> )</span><br/>

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c14">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">n</font>,<font color="Maroon" title="c9">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">E</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c15">x'</font> = <font color="Maroon" title="c14">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">X</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E12:21"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E3:21_1"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c11">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">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="E4:21_1"/>
<font color="Maroon" title="c2">k</font> <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#E2:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="card_1.html#T46" title="CARD_1:th.46">CARD_1:46</a></i>;<br/>
<a NAME="E5:21_1"/><b>then </b><i><font color="Green" title="E26">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">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> <font color="Maroon" title="c1">n</font>,<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="E6:21_1"/>
<font color="Maroon" title="c15">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="c8">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E5:21_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="ramsey_1.html#E1:21_1"><i><font color="Green" title="E24">A7</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="E7:21_1"/><b>then </b><i><font color="Green" title="E27">A10</font></i>: 
<font color="Maroon" title="c15">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="c8">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</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="E8:21_1"/>
<font color="Maroon" title="c11">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="c8">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E5:21_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="ramsey_1.html#E15:21"><i><font color="Green" title="E23">A6</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="E9:21_1"/><b>then </b><i><font color="Green" title="E28">A11</font></i>: 
<font color="Maroon" title="c11">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="c8">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</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="E29">A12</font></i>: <font color="Maroon" title="c8">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x'</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E1:21_1"><i><font color="Green" title="E24">A7</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="c8">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">h</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E9:21"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="ramsey_1.html#E7:21_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="txt" href="ramsey_1.html#E9:21_1"><i><font color="Green" title="E28">A11</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="c8">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">h</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E15:21"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<font color="Maroon" title="c6">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="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x'</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c4">P</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">h</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E10:21_1"><i><font color="Green" title="E29">A12</font></i></a>, <a class="txt" href="ramsey_1.html#E5:21_1"><i><font color="Green" title="E26">A9</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="c6">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="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">h</font></span>)</span>
<b>by </b><i><a class="txt" href="ramsey_1.html#E5:21_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/>
<a NAME="E12:21_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">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="c4">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x'</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E4:21"><i><font color="Green" title="E19">A2</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="E13:21_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">E</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:21_1"><i><font color="Green" title="E25">A8</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="E20:21"/><b>then </b>
 <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">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="E21:21"/>
<font color="Maroon" title="c9">H</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>
