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

<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="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> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c2">k</font>  st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <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>  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="b2">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="Olive" title="b3">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/><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"> for <font color="Olive" title="b1">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">k</font>  st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <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>  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="b1">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="Olive" title="b2">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b>for <font color="Olive" title="b1">k</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/> for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">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> $1,<font color="Olive" title="b2">X</font>,<font color="Olive" title="b1">k</font>  st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b1">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>  holds <br/> ex <font color="Olive" title="b4">H</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/>( not <font color="Olive" title="b4">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">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> $1,<font color="Olive" title="b4">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> );<br/>
<a NAME="E1:14"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c4">k</font>  st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c4">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>  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="b2">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Olive" title="b3">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/><b>let </b><font color="Maroon" title="c5">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"> for <font color="Olive" title="b1">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c4">k</font>  st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c4">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>  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="c5">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="b1">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Olive" title="b2">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/><b>let </b><font color="Maroon" title="c6">F</font> be   <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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c4">k</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="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c4">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>  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="c5">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="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Olive" title="b1">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> ) )</span><br/>





<b>assume </b><a NAME="E1:14_1"/><i><font color="Green" title="E13">A2</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  or  not <font color="Maroon" title="c4">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>  or  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="c5">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="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Olive" title="b1">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> ) )</span><br/>

<b>assume </b><a NAME="E2:14_1"/>
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">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>  or  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="c5">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="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Olive" title="b1">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> ) )</span><br/>

<b>assume </b><a NAME="E3:14_1"/>
<font color="Maroon" title="c4">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> 
 ; <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="c5">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="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Olive" title="b1">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/>

<b>set </b><font color="Maroon" title="c7">H</font> = <font color="Maroon" title="c5">X</font>;<br/>
<a NAME="E4:14_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">X</font>
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c8">H</font> = <font color="Maroon" title="c5">X</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c5">X</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_1"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">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="c8">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/>

<b>thus </b><a NAME="E6:14_1"/>
not <font color="Maroon" title="c8">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#E1:14_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</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">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a></span><br/>

<a NAME="E7:14_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</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="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</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="c9">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="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">y</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="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> )</span><br/>

<i><font color="Green" title="E15">A4</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="ramsey_1.html#E5"><i><font color="Green" title="E5">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
;<br/>
<b>assume </b><a NAME="E2:14_1_1"/>
<font color="Maroon" title="c9">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="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">y</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="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> )</span><br/>

<a NAME="E3:14_1_1"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>assume </b><a NAME="E4:14_1_1"/>
<font color="Maroon" title="c10">y</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="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font></span><br/>

<a NAME="E5:14_1_1"/><b>then </b>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_1_1"><i><font color="Green" title="E15">A4</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="E6:14_1_1"/><b>then </b>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:14_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <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><b>end;</b></div>
<b>hence </b><a NAME="E8:14_1"/>
<font color="Maroon" title="c6">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> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c8">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</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="E2:14"/><i><font color="Green" title="E13">A6</font></i>: 
for <font color="Olive" title="b1">n</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">n</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:14_2"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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="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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c5">k</font>  st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c5">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>  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="b2">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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Olive" title="b3">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/><b>let </b><font color="Maroon" title="c6">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"> for <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c5">k</font>  st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c6">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c5">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>  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="c6">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="b1">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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Olive" title="b2">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/><b>let </b><font color="Maroon" title="c7">F</font> be   <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c5">k</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="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c6">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c5">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>  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="c6">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="Maroon" title="c7">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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Olive" title="b1">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> ) )</span><br/><b>assume </b><a NAME="E1:14_2_1"/><i><font color="Green" title="E15">A8</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c5">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>  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="c6">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="Maroon" title="c7">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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Olive" title="b1">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> ) )</span><br/><b>assume </b><a NAME="E2:14_2_1"/><i><font color="Green" title="E16">A9</font></i>: 
<font color="Maroon" title="c6">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">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>  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="c6">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="Maroon" title="c7">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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Olive" title="b1">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> ) )</span><br/><b>assume </b><a NAME="E3:14_2_1"/><i><font color="Green" title="E17">A10</font></i>: 
<font color="Maroon" title="c5">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> 
 ; <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="c6">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="Maroon" title="c7">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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Olive" title="b1">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/><a NAME="E4:14_2_1"/><i><font color="Green" title="E18">A11</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">Y</font>  st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> holds <br/> ex <font color="Olive" title="b3">Fa</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b4">Ha</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b4">Ha</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">Fa</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="c4">n</font>,<font color="Olive" title="b4">Ha</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; ( for <font color="Olive" title="b5">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b3">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">Y</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"> for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">Y</font>  st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> holds <br/> ex <font color="Olive" title="b2">Fa</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b3">Ha</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b3">Ha</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b2">Fa</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="c4">n</font>,<font color="Olive" title="b3">Ha</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; ( for <font color="Olive" title="b4">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b2">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c9">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">Y</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="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> implies  ex <font color="Olive" title="b1">Fa</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b2">Ha</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b2">Ha</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">Fa</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="c4">n</font>,<font color="Olive" title="b2">Ha</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; ( for <font color="Olive" title="b3">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) ) )</span><br/>



<b>assume </b><a NAME="E1:14_2_1_1"/><i><font color="Green" title="E19">A12</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font> or  ex <font color="Olive" title="b1">Fa</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b2">Ha</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b2">Ha</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">Fa</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="c4">n</font>,<font color="Olive" title="b2">Ha</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; ( for <font color="Olive" title="b3">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) ) )</span><br/>

<b>assume </b><a NAME="E2:14_2_1_1"/><i><font color="Green" title="E20">A13</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</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">Fa</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b2">Ha</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b2">Ha</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">Fa</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="c4">n</font>,<font color="Olive" title="b2">Ha</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; ( for <font color="Olive" title="b3">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) )</span><br/>

<b>set </b><font color="Maroon" title="c10">Y1</font> =  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<a NAME="E3:14_2_1_1"/><i><font color="Green" title="E21">A14</font></i>: 
not <font color="Maroon" title="c8">Y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_1"><i><font color="Green" title="E19">A12</font></i></a></i>;<br/>
<a NAME="E4:14_2_1_1"/>
not <font color="Maroon" title="c8">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_1"><i><font color="Green" title="E19">A12</font></i></a></i>;<br/>
<a NAME="E5:14_2_1_1"/><b>then </b><i><font color="Green" title="E22">A15</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E6:14_2_1_1"/><i><font color="Green" title="E23">A16</font></i>: 
not <font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_1_1"/><i><font color="Green" title="E24">A17</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> 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#E1:14_2_1_1_1"><i><font color="Green" title="E24">A17</font></i></a>, <a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">FINSET_1:14</a></i>;<br/>
<a NAME="E3:14_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:14_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_1"><i><font color="Green" title="E21">A14</font></i></a>, <a class="txt" href="ramsey_1.html#E5:14_2_1_1"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</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="E7:14_2_1_1"/>
not  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> 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#E6:14_2_1_1"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a></i>;<br/>
<a NAME="E8:14_2_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_4.html#T13" title="CARD_4:th.13">CARD_4:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">Y1</font> =  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> 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#E6:14_2_1_1"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="group_10.html#T2" title="GROUP_10:th.2">GROUP_10:2</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">Y1</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<a NAME="E10:14_2_1_1"/><i><font color="Green" title="E24">A18</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">Y1</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">Y1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font></span><br/>

<a NAME="E1:14_2_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y1</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c13">x'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span><b> such that </b><br/><a NAME="E3:14_2_1_1_2"/><i><font color="Green" title="E25">A19</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x'</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/>
<a NAME="E4:14_2_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_1_2"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E5:14_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c12">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</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="c8">Y</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E5:14_2_1_1"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x''</font> = <font color="Maroon" title="c13">x'</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_1_2"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a></i>;<br/>
<a NAME="E8:14_2_1_1_2"/><i><font color="Green" title="E26">A20</font></i>: 
not <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">x''</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_1_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">x''</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_2_1_1_2_1"/><b>then </b>
not <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:14_2_1_1_2_1"/>
contradiction
 <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><b>end;</b></div>
<a NAME="E9:14_2_1_1_2"/><i><font color="Green" title="E27">A21</font></i>: 
 <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c8">Y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c6">X</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_1"><i><font color="Green" title="E20">A13</font></i></a>, <a class="txt" href="ramsey_1.html#E3"><i><font color="Green" title="E3">Lm1</font></i></a></i>;<br/>
<a NAME="E10:14_2_1_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_1_2"><i><font color="Green" title="E25">A19</font></i></a>, <a class="txt" href="ramsey_1.html#E8:14_2_1_1_2"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/>
<a NAME="E11:14_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c8">Y</font>
 
;<br/>
<b>hence </b><a NAME="E12:14_2_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="ramsey_1.html#E9:14_2_1_1_2"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</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>consider </b><font color="Maroon" title="c12">Fa</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c11">Y1</font>,<font color="Maroon" title="c5">k</font><b> such that </b><br/><a NAME="E12:14_2_1_1"/><i><font color="Green" title="E25">A22</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">Y1</font> holds  <font color="Maroon" title="c12">Fa</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="ramsey_1.html#E10:14_2_1_1"><i><font color="Green" title="E24">A18</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c13">Fa</font> = <font color="Maroon" title="c12">Fa</font> 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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ;<br/>
<b>set </b><font color="Maroon" title="c14">Y'</font> = <font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E14:14_2_1_1"/><i><font color="Green" title="E26">A23</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="ramsey_1.html#E2:14_2_1_1"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:14_2_1_1"/><b>then </b><i><font color="Green" title="E27">A24</font></i>: 
<font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:14_2_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E6:14_2_1_1"><i><font color="Green" title="E23">A16</font></i></a>, <a class="txt" href="ramsey_1.html#E14:14_2_1_1"><i><font color="Green" title="E26">A23</font></i></a>, <a class="ref" href="ramsey_1.html#T2" target="_self" title="RAMSEY_1:th.2">Th2</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">Ha</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span><b> such that </b><br/><a NAME="E18:14_2_1_1"/><i><font color="Green" title="E28">A25</font></i>: 
( not <font color="Maroon" title="c15">Ha</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c13">Fa</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="c4">n</font>,<font color="Maroon" title="c15">Ha</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#E1:14_2"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="ramsey_1.html#E3:14_2_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="ramsey_1.html#E15:14_2_1_1"><i><font color="Green" title="E27">A24</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">Fa</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">Ha</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b1">Ha</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c13">Fa</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="c4">n</font>,<font color="Olive" title="b1">Ha</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; ( for <font color="Olive" title="b2">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c13">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c15">Ha</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="c15">Ha</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c13">Fa</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="c4">n</font>,<font color="Maroon" title="c15">Ha</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; ( for <font color="Olive" title="b1">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c13">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) )</span><br/>

<b>thus </b><a NAME="E19:14_2_1_1"/>
( not <font color="Maroon" title="c15">Ha</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c13">Fa</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="c4">n</font>,<font color="Maroon" title="c15">Ha</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#E18:14_2_1_1"><i><font color="Green" title="E28">A25</font></i></a></i>; <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">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c13">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c16">Y'</font> be    <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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/>

<b>thus </b><a NAME="E20:14_2_1_1"/>
<font color="Maroon" title="c13">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E12:14_2_1_1"><i><font color="Green" title="E25">A22</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><a NAME="E5:14_2_1"/>
<font color="Maroon" title="c6">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font>
 ;<br/><a NAME="E6:14_2_1"/><b>then </b><i><font color="Green" title="E19">A26</font></i>: 
<font color="Maroon" title="c6">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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c6">X</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1"><i><font color="Green" title="E15">A8</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">A</font> =  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c6">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>  ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font><br/> for <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y1</font> &amp; $4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font> &amp; $5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y2</font> holds <br/>( ( <font color="Olive" title="b3">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x1</font> implies  ex <font color="Olive" title="b5">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b6">H1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b6">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b5">F1</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="c4">n</font>,<font color="Olive" title="b6">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">H1</font> &amp; <font color="Olive" title="b4">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">H1</font> &amp; <font color="Olive" title="b3">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">y2</font> &amp; ( for <font color="Olive" title="b7">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b7">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b7">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b3">y1</font> &amp; <font color="Olive" title="b7">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">H1</font> holds <br/><font color="Olive" title="b4">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b7">y2'</font> ) ) ) &amp; ( not <font color="Olive" title="b3">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x1</font> implies ( <font color="Olive" title="b2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Olive" title="b4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) ) );<br/><a NAME="E8:14_2_1"/><i><font color="Green" title="E20">A27</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font><br/> for <font color="Olive" title="b3">y''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   ex <font color="Olive" title="b4">x1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> ex <font color="Olive" title="b5">y1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x''</font>,<font color="Olive" title="b3">y''</font>,<font color="Olive" title="b4">x1''</font>,<font color="Olive" title="b5">y1''</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font><br/> for <font color="Olive" title="b2">y''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   ex <font color="Olive" title="b3">x1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> ex <font color="Olive" title="b4">y1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Olive" title="b1">x''</font>,<font color="Olive" title="b2">y''</font>,<font color="Olive" title="b3">x1''</font>,<font color="Olive" title="b4">y1''</font>]</span><br/><b>let </b><font color="Maroon" title="c10">x''</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</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">y''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   ex <font color="Olive" title="b2">x1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> ex <font color="Olive" title="b3">y1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Olive" title="b1">y''</font>,<font color="Olive" title="b2">x1''</font>,<font color="Olive" title="b3">y1''</font>]</span><br/><b>let </b><font color="Maroon" title="c11">y''</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</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">x1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> ex <font color="Olive" title="b2">y1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Olive" title="b1">x1''</font>,<font color="Olive" title="b2">y1''</font>]</span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1_2_1"/>
( <font color="Maroon" title="c11">y''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x''</font> or  not <font color="Maroon" title="c11">y''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x''</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_2_1_1"/><i><font color="Green" title="E21">A28</font></i>: 
<font color="Maroon" title="c11">y''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x''</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">x1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> ex <font color="Olive" title="b2">y1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Olive" title="b1">x1''</font>,<font color="Olive" title="b2">y1''</font>]</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c12">a'</font> = <font color="Maroon" title="c11">y''</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">x''</font> ;<br/><a NAME="E3:14_2_1_2_1_1"/>
<font color="Maroon" title="c10">x''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">x'''</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c6">X</font><b> such that </b><br/><a NAME="E5:14_2_1_2_1_1"/><i><font color="Green" title="E22">A29</font></i>: 
( <font color="Maroon" title="c13">x'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x''</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">x'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 ;<br/><b>consider </b><font color="Maroon" title="c14">Fa</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">x''</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>, <font color="Maroon" title="c15">Ha</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c10">x''</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span><b> such that </b><br/><a NAME="E7:14_2_1_2_1_1"/><i><font color="Green" title="E23">A30</font></i>: 
( not <font color="Maroon" title="c15">Ha</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c14">Fa</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="c4">n</font>,<font color="Maroon" title="c15">Ha</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; ( for <font color="Olive" title="b1">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">x''</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c14">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="ramsey_1.html#E5:14_2_1_2_1_1"><i><font color="Green" title="E22">A29</font></i></a></i>;<br/><a NAME="E8:14_2_1_2_1_1"/>
<font color="Maroon" title="c15">Ha</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">x''</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:14_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E24">A31</font></i>: 
<font color="Maroon" title="c15">Ha</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E5:14_2_1_2_1_1"><i><font color="Green" title="E22">A29</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:14_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E25">A32</font></i>: 
<font color="Maroon" title="c15">Ha</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:14_2_1_2_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c15">Ha</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="ramsey_1.html#E7:14_2_1_2_1_1"><i><font color="Green" title="E23">A30</font></i></a>, <a class="ref" href="ramsey_1.html#T2" target="_self" title="RAMSEY_1:th.2">Th2</a></i>;<br/><a NAME="E12:14_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c15">Ha</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E9:14_2_1_2_1_1"><i><font color="Green" title="E24">A31</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">x1''</font> = <font color="Maroon" title="c15">Ha</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> ;<br/><b>set </b><font color="Maroon" title="c17">y1''</font> =  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ;<br/><b>take </b>
<font color="Maroon" title="c16">x1''</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">y1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c16">x1''</font>,<font color="Olive" title="b1">y1''</font>]</span><br/><b>take </b>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c16">x1''</font>, <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">x1</font>, <font color="Maroon" title="c19">x2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</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">y1</font>, <font color="Olive" title="b2">y2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Maroon" title="c10">x''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x1</font> &amp; <font color="Maroon" title="c11">y''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y1</font> &amp; <font color="Maroon" title="c16">x1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x2</font> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">y2'</font> where <font color="Olive" title="b3">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b3">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b3">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y2</font> holds <br/>( ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font> implies  ex <font color="Olive" title="b3">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b4">H1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b4">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b3">F1</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="c4">n</font>,<font color="Olive" title="b4">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">H1</font> &amp; <font color="Olive" title="b2">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">H1</font> &amp; <font color="Olive" title="b1">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">y2</font> &amp; ( for <font color="Olive" title="b5">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b5">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b5">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">y1</font> &amp; <font color="Olive" title="b5">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">H1</font> holds <br/><font color="Olive" title="b2">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">y2'</font> ) ) ) &amp; ( not <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font> implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c16">x1''</font>, <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">y2'</font> where <font color="Olive" title="b3">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b3">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b3">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ] ) )</span><br/><b>let </b><font color="Maroon" title="c20">y1</font>, <font color="Maroon" title="c21">y2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x1</font> &amp; <font color="Maroon" title="c11">y''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Maroon" title="c16">x1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x2</font> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y2</font> implies ( ( <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font> implies  ex <font color="Olive" title="b1">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b2">H1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b2">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">F1</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="c4">n</font>,<font color="Olive" title="b2">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font> &amp; ( for <font color="Olive" title="b3">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b3">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b3">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">y2'</font> ) ) ) &amp; ( not <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font> implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c16">x1''</font>, <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ] ) ) )</span><br/><b>assume </b><a NAME="E1:14_2_1_2_1_1_1"/><i><font color="Green" title="E26">A33</font></i>: 
( <font color="Maroon" title="c10">x''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x1</font> &amp; <font color="Maroon" title="c11">y''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y1</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">x1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x2</font> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y2</font> implies ( ( <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font> implies  ex <font color="Olive" title="b1">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b2">H1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b2">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">F1</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="c4">n</font>,<font color="Olive" title="b2">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font> &amp; ( for <font color="Olive" title="b3">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b3">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b3">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">y2'</font> ) ) ) &amp; ( not <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font> implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c16">x1''</font>, <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ] ) ) )</span><br/><b>assume </b><a NAME="E2:14_2_1_2_1_1_1"/><i><font color="Green" title="E27">A34</font></i>: 
( <font color="Maroon" title="c16">x1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x2</font> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font> implies  ex <font color="Olive" title="b1">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b2">H1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b2">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">F1</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="c4">n</font>,<font color="Olive" title="b2">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font> &amp; ( for <font color="Olive" title="b3">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b3">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b3">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">y2'</font> ) ) ) &amp; ( not <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font> implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c16">x1''</font>, <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ] ) )</span><br/><b>thus </b><a NAME="E3:14_2_1_2_1_1_1"/>
( <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font> implies  ex <font color="Olive" title="b1">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b2">H1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b2">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">F1</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="c4">n</font>,<font color="Olive" title="b2">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font> &amp; ( for <font color="Olive" title="b3">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b3">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b3">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">y2'</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="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font> implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c16">x1''</font>, <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ] )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</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">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b2">H1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b2">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">F1</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="c4">n</font>,<font color="Olive" title="b2">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font> &amp; ( for <font color="Olive" title="b3">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b3">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b3">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">y2'</font> ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c22">F1</font> = <font color="Maroon" title="c14">Fa</font> 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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_1_1"><i><font color="Green" title="E26">A33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">H1</font> = <font color="Maroon" title="c15">Ha</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_1_1"><i><font color="Green" title="E26">A33</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c22">F1</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">H1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b1">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c22">F1</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="c4">n</font>,<font color="Olive" title="b1">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> &amp; <font color="Maroon" title="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font> &amp; ( for <font color="Olive" title="b2">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c22">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b2">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b2">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b2">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">y2'</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c23">H1</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="c23">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c22">F1</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="c4">n</font>,<font color="Maroon" title="c23">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">H1</font> &amp; <font color="Maroon" title="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font> &amp; ( for <font color="Olive" title="b1">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c22">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y2'</font> ) )</span><br/>

<b>thus </b><a NAME="E4:14_2_1_2_1_1_1_1"/>
not <font color="Maroon" title="c23">H1</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#E7:14_2_1_2_1_1"><i><font color="Green" title="E23">A30</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="c22">F1</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="c4">n</font>,<font color="Maroon" title="c23">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">H1</font> &amp; <font color="Maroon" title="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font> &amp; ( for <font color="Olive" title="b1">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c22">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y2'</font> ) )</span><br/>

<b>thus </b><a NAME="E5:14_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c22">F1</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="c4">n</font>,<font color="Maroon" title="c23">H1</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#E7:14_2_1_2_1_1"><i><font color="Green" title="E23">A30</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="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">H1</font> &amp; <font color="Maroon" title="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font> &amp; ( for <font color="Olive" title="b1">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c22">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y2'</font> ) )</span><br/>

<b>thus </b><a NAME="E6:14_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c19">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">H1</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_2_1_1_1"><i><font color="Green" title="E27">A34</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="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font> &amp; ( for <font color="Olive" title="b1">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c22">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y2'</font> ) )</span><br/>

<b>set </b><font color="Maroon" title="c24">A'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ;<br/>
<a NAME="E7:14_2_1_2_1_1_1_1"/>
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> <span class="p1">{<span class="default"> <font color="Olive" title="b2">y2'</font> where <font color="Olive" title="b2">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b2">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b2">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">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="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  implies <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<b>assume </b><a NAME="E1:14_2_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c26">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> <b> such that </b><br/><a NAME="E3:14_2_1_2_1_1_1_1_1"/><i><font color="Green" title="E28">A35</font></i>: 
( <font color="Maroon" title="c26">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c26">x'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Maroon" title="c26">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> )
 ;<br/>
<b>thus </b><a NAME="E4:14_2_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_2_1_1_1_1_1"><i><font color="Green" title="E28">A35</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>then reconsider </b><font color="Maroon" title="c25">A'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E9:14_2_1_2_1_1_1_1"/><i><font color="Green" title="E28">A36</font></i>: 
<font color="Maroon" title="c25">A'</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_2_1_1_1_1_2"/><i><font color="Green" title="E29">A37</font></i>: 
<font color="Maroon" title="c25">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>set </b><font color="Maroon" title="c26">A''</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">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="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  implies <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y''</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> )</span><br/><b>assume </b><a NAME="E1:14_2_1_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y''</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c28">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> <b> such that </b><br/><a NAME="E3:14_2_1_2_1_1_1_1_2_1"/><i><font color="Green" title="E30">A38</font></i>: 
( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">x'</font> &amp; <font color="Maroon" title="c28">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Maroon" title="c28">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> )
 ;<br/><a NAME="E4:14_2_1_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c28">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">y''</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_2_1_1_1_1_2_1"><i><font color="Green" title="E30">A38</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E5:14_2_1_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y''</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a>, <a class="txt" href="ramsey_1.html#E3:14_2_1_2_1_1_1_1_2_1"><i><font color="Green" title="E30">A38</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>
<a NAME="E3:14_2_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y''</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:14_2_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E30">A39</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<div><i><font color="Green" title="E31">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">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="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> )</span><br/><b>assume </b><a NAME="E1:14_2_1_2_1_1_1_1_2_2"/><i><font color="Green" title="E32">A41</font></i>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1_2_1_1_1_1_2_2_1"/>
( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font> or <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_1_1_1_2_2"><i><font color="Green" title="E32">A41</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2_1_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_2_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c28">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> <b> such that </b><br/><a NAME="E3:14_2_1_2_1_1_1_1_2_2_1_1"/><i><font color="Green" title="E33">A42</font></i>: 
( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">x'</font> &amp; <font color="Maroon" title="c28">x'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Maroon" title="c28">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> )
 ;<br/><b>thus </b><a NAME="E4:14_2_1_2_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_2_1_1_1_1_2_2_1_1"><i><font color="Green" title="E33">A42</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2_1_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_2_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c28">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> <b> such that </b><br/><a NAME="E3:14_2_1_2_1_1_1_1_2_2_1_2"/><i><font color="Green" title="E33">A43</font></i>: 
( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">x'</font> &amp; <font color="Maroon" title="c28">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Maroon" title="c28">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> )
 ;<br/><b>thus </b><a NAME="E4:14_2_1_2_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_2_1_1_1_1_2_2_1_2"><i><font color="Green" title="E33">A43</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></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_1_1_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">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="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  )</span><br/><b>assume </b><a NAME="E1:14_2_1_2_1_1_1_1_2_3"/><i><font color="Green" title="E32">A44</font></i>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> </span><br/><b>then reconsider </b><font color="Maroon" title="c28">x'</font> = <font color="Maroon" title="c27">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="txt" href="ramsey_1.html#E10:14_2_1_2_1_1"><i><font color="Green" title="E25">A32</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2_1_1_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1_2_1_1_1_1_2_3_1"/>
( <font color="Maroon" title="c28">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> or <font color="Maroon" title="c28">x'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2_1_1_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_2_1_1_1_1_2_3_1_1"/>
<font color="Maroon" title="c28">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> </span><br/><div class="add"><a NAME="E2:14_2_1_2_1_1_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_1_1_1_2_3"><i><font color="Green" title="E32">A44</font></i></a></i>;<br/><b>hence </b><a NAME="E3:14_2_1_2_1_1_1_1_2_3_1_1"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2_1_1_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_2_1_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c28">x'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> </span><br/><div class="add"><a NAME="E2:14_2_1_2_1_1_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_1_1_1_2_3"><i><font color="Green" title="E32">A44</font></i></a></i>;<br/><b>hence </b><a NAME="E3:14_2_1_2_1_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'''</font> where <font color="Olive" title="b1">y2'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'''</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:14_2_1_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="ramsey_1.html#E7:14_2_1_2_1_1"><i><font color="Green" title="E23">A30</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_1_1_1_2"><i><font color="Green" title="E29">A37</font></i></a>, <a class="txt" href="ramsey_1.html#E4:14_2_1_2_1_1_1_1_2"><i><font color="Green" title="E30">A39</font></i></a>, <a class="txt" href="ramsey_1.html#E5:14_2_1_2_1_1_1_1_2"><i><font color="Green" title="E31">A40</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI: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>
<a NAME="E10:14_2_1_2_1_1_1_1"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E9:14_2_1_2_1_1_1_1"><i><font color="Green" title="E28">A36</font></i></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c26">y2''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> <b> such that </b><br/><a NAME="E12:14_2_1_2_1_1_1_1"/><i><font color="Green" title="E29">A45</font></i>: 
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">y2''</font> &amp; <font color="Maroon" title="c26">y2''</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Maroon" title="c26">y2''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> )
 ;<br/>
<b>thus </b><a NAME="E13:14_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c21">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E12:14_2_1_2_1_1_1_1"><i><font color="Green" title="E29">A45</font></i></a>, <a class="txt" href="ramsey_1.html#E2:14_2_1_2_1_1_1"><i><font color="Green" title="E27">A34</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="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font> &amp; ( for <font color="Olive" title="b1">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c22">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y2'</font> ) )</span><br/>

<b>thus </b><a NAME="E14:14_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c20">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">y2</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E12:14_2_1_2_1_1_1_1"><i><font color="Green" title="E29">A45</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_1_1"><i><font color="Green" title="E26">A33</font></i></a>, <a class="txt" href="ramsey_1.html#E2:14_2_1_2_1_1_1"><i><font color="Green" title="E27">A34</font></i></a></i>; <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">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c22">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y2'</font> ) )</span><br/>

<b>thus </b><a NAME="E15:14_2_1_2_1_1_1_1"/>
for <font color="Olive" title="b1">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c22">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_1_1"><i><font color="Green" title="E26">A33</font></i></a>, <a class="txt" href="ramsey_1.html#E7:14_2_1_2_1_1"><i><font color="Green" title="E23">A30</font></i></a></i>; <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">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> holds <br/><font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y2'</font></span><br/>

<b>let </b><font color="Maroon" title="c27">y2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c27">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</font> &amp; <font color="Maroon" title="c27">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> implies <font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">y2'</font> )</span><br/>

<b>assume </b><a NAME="E16:14_2_1_2_1_1_1_1"/><i><font color="Green" title="E30">A46</font></i>: 
<font color="Maroon" title="c27">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">y1</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="c27">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font> or <font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">y2'</font> )</span><br/>

<b>assume </b><a NAME="E17:14_2_1_2_1_1_1_1"/><i><font color="Green" title="E31">A47</font></i>: 
<font color="Maroon" title="c27">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">y2'</font></span><br/>

<a NAME="E18:14_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c27">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">A'</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E16:14_2_1_2_1_1_1_1"><i><font color="Green" title="E30">A46</font></i></a>, <a class="txt" href="ramsey_1.html#E17:14_2_1_2_1_1_1_1"><i><font color="Green" title="E31">A47</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_1_1"><i><font color="Green" title="E26">A33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:14_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c21">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">y2'</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_2_1_1_1"><i><font color="Green" title="E27">A34</font></i></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1: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><b>end;</b></div><b>assume </b><a NAME="E4:14_2_1_2_1_1_1"/>
not <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c16">x1''</font>, <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ]</span><br/><b>hence </b><a NAME="E5:14_2_1_2_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c16">x1''</font>, <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ]
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_1_1"><i><font color="Green" title="E26">A33</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_1"><i><font color="Green" title="E21">A28</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>hence </b><a NAME="E15:14_2_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c16">x1''</font>, <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y2'</font> where <font color="Olive" title="b1">y2'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  : ( <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">y''</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Ha</font> ) </span>}</span> ]
 ; <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="14_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_2_1_2"/><i><font color="Green" title="E21">A48</font></i>: 
not <font color="Maroon" title="c11">y''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">x''</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">x1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> ex <font color="Olive" title="b2">y1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Olive" title="b1">x1''</font>,<font color="Olive" title="b2">y1''</font>]</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c12">x1''</font> = <font color="Maroon" title="c6">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E6:14_2_1"><i><font color="Green" title="E19">A26</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">y1''</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><b>take </b>
<font color="Maroon" title="c12">x1''</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">y1''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c12">x1''</font>,<font color="Olive" title="b1">y1''</font>]</span><br/><b>take </b>
<font color="Maroon" title="c13">y1''</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c12">x1''</font>,<font color="Maroon" title="c13">y1''</font>]</span><br/><b>thus </b><a NAME="E4:14_2_1_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">x''</font>,<font color="Maroon" title="c11">y''</font>,<font color="Maroon" title="c12">x1''</font>,<font color="Maroon" title="c13">y1''</font>]
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_2_1_2"><i><font color="Green" title="E21">A48</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></div><b>end;</b></div>

</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c9">X0</font> = <font color="Maroon" title="c6">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E6:14_2_1"><i><font color="Green" title="E19">A26</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c10">Y0</font> =  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c6">X</font>;<br/><b>consider </b><font color="Maroon" title="c11">S</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c8">A</font>, <font color="Maroon" title="c12">a</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> <b> such that </b><br/><a NAME="E11:14_2_1"/><i><font color="Green" title="E21">A49</font></i>: 
( <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">X0</font> &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c6">X</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_2.html#S3" title="RECDEF_2:sch.3">RECDEF_2:sch 3</a>(<a class="txt" href="ramsey_1.html#E8:14_2_1"><i><font color="Green" title="E20">A27</font></i></a>);<br/></i><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b>( <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default">$1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default">$1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default">$1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default">$1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; not <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default">$1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> );<br/><a NAME="E12:14_2_1"/><i><font color="Green" title="E22">A50</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c13">i</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">i</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x1</font> = <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">x2</font> = <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">y1</font> = <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c18">y2</font> = <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  ;<br/>
<a NAME="E6:14_2_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E11:14_2_1"><i><font color="Green" title="E21">A49</font></i></a></i>;<br/>
<a NAME="E7:14_2_1_3"/><b>then </b><i><font color="Green" title="E23">A51</font></i>: 
( ( <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">x1</font> implies  ex <font color="Olive" title="b1">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b2">H1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b2">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">F1</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="c4">n</font>,<font color="Olive" title="b2">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c18">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c17">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">y2</font> &amp; ( for <font color="Olive" title="b3">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b3">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">y1</font> &amp; <font color="Olive" title="b3">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> holds <br/><font color="Maroon" title="c18">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">y2'</font> ) ) ) &amp; ( not <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">x1</font> implies ( <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c18">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) ) )
 
;<br/>
<b>consider </b><font color="Maroon" title="c19">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>, <font color="Maroon" title="c20">H1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span><b> such that </b><br/><a NAME="E9:14_2_1_3"/><i><font color="Green" title="E24">A52</font></i>: 
( not <font color="Maroon" title="c20">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c19">F1</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="c4">n</font>,<font color="Maroon" title="c20">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">H1</font> &amp; <font color="Maroon" title="c18">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">H1</font> &amp; <font color="Maroon" title="c17">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">y2</font> )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E7:14_2_1_3"><i><font color="Green" title="E23">A51</font></i></a>, <a class="txt" href="ramsey_1.html#E11:14_2_1"><i><font color="Green" title="E21">A49</font></i></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a>, <a class="txt" href="ramsey_1.html#E1:14_2_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="ramsey_1.html#E2:14_2_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<b>thus </b><a NAME="E10:14_2_1_3"/>
<font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E9:14_2_1_3"><i><font color="Green" title="E24">A52</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; not <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> )</span><br/>

<b>thus </b><a NAME="E11:14_2_1_3"/>
<font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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#E9:14_2_1_3"><i><font color="Green" title="E24">A52</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; not <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> )</span><br/>

<b>thus </b><a NAME="E12:14_2_1_3"/>
<font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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#E11:14_2_1"><i><font color="Green" title="E21">A49</font></i></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a>, <a class="txt" href="ramsey_1.html#E1:14_2_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="ramsey_1.html#E2:14_2_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</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="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; not <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> )</span><br/>

<b>thus </b><a NAME="E13:14_2_1_3"/>
<font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E9:14_2_1_3"><i><font color="Green" title="E24">A52</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span><br/>

<a NAME="E14:14_2_1_3"/>
not <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_3_1"/>
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_2_1_3_1"/><b>then </b>
not <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E9:14_2_1_3"><i><font color="Green" title="E24">A52</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:14_2_1_3_1"/>
contradiction
 <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><b>end;</b></div>
<b>hence </b><a NAME="E15:14_2_1_3"/>
not <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 ; <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="E13:14_2_1"/><i><font color="Green" title="E23">A53</font></i>: 
for <font color="Olive" title="b1">i</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 <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</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"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">i</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:14_2_1_4"/><i><font color="Green" title="E24">A54</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<b>reconsider </b><font color="Maroon" title="c14">i'</font> = <font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x1</font> = <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">x2</font> = <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">y1</font> = <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c18">y2</font> = <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c19">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>, <font color="Maroon" title="c20">H1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span><b> such that </b><br/><a NAME="E8:14_2_1_4"/><i><font color="Green" title="E25">A55</font></i>: 
( not <font color="Maroon" title="c20">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c19">F1</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="c4">n</font>,<font color="Maroon" title="c20">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">H1</font> &amp; <font color="Maroon" title="c18">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">H1</font> &amp; <font color="Maroon" title="c17">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">y2</font> &amp; ( for <font color="Olive" title="b1">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">y1</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">H1</font> holds <br/><font color="Maroon" title="c18">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y2'</font> ) )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E11:14_2_1"><i><font color="Green" title="E21">A49</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_4"><i><font color="Green" title="E24">A54</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:14_2_1_4"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:14_2_1_4_1"/>
<font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E8:14_2_1_4"><i><font color="Green" title="E25">A55</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; not <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> )</span><br/>

<b>thus </b><a NAME="E2:14_2_1_4_1"/>
<font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E8:14_2_1_4"><i><font color="Green" title="E25">A55</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; not <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> )</span><br/>

<b>thus </b><a NAME="E3:14_2_1_4_1"/>
<font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_4"><i><font color="Green" title="E24">A54</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="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; not <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> )</span><br/>

<b>thus </b><a NAME="E4:14_2_1_4_1"/>
<font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E8:14_2_1_4"><i><font color="Green" title="E25">A55</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span><br/>

<a NAME="E5:14_2_1_4_1"/>
not <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_4_1_1"/>
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_2_1_4_1_1"/><b>then </b>
not <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E8:14_2_1_4"><i><font color="Green" title="E25">A55</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:14_2_1_4_1_1"/>
contradiction
 <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><b>end;</b></div>
<b>hence </b><a NAME="E6:14_2_1_4_1"/>
not <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 ; <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><a NAME="E14:14_2_1"/><i><font color="Green" title="E24">A56</font></i>: 
for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="ramsey_1.html#E12:14_2_1"><i><font color="Green" title="E22">A50</font></i></a>, <a class="txt" href="ramsey_1.html#E13:14_2_1"><i><font color="Green" title="E23">A53</font></i></a>);<br/></i><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b>for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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 <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">j</font> holds <br/>( <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">j</font> &amp; ( for <font color="Olive" title="b3">ai</font>, <font color="Olive" title="b4">aj</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 <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b3">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Olive" title="b3">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b4">aj</font> ) );<br/><a NAME="E15:14_2_1"/><i><font color="Green" title="E25">A57</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 ;<br/><a NAME="E16:14_2_1"/><i><font color="Green" title="E26">A58</font></i>: 
for <font color="Olive" title="b1">l</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 <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">l</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">l</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"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">l</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:14_2_1_5"/><i><font color="Green" title="E27">A59</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">l</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<b>thus </b><a NAME="E2:14_2_1_5"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font>, <font color="Maroon" title="c15">j</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"> ( <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">j</font> implies ( <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> &amp; ( for <font color="Olive" title="b1">ai</font>, <font color="Olive" title="b2">aj</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 <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Olive" title="b1">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Olive" title="b2">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> holds <br/><font color="Olive" title="b1">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">aj</font> ) ) )</span><br/>

<b>assume </b><a NAME="E1:14_2_1_5_1"/><i><font color="Green" title="E28">A60</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">j</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="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">j</font> or ( <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> &amp; ( for <font color="Olive" title="b1">ai</font>, <font color="Olive" title="b2">aj</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 <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Olive" title="b1">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Olive" title="b2">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> holds <br/><font color="Olive" title="b1">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">aj</font> ) ) )</span><br/>

<b>assume </b><a NAME="E2:14_2_1_5_1"/><i><font color="Green" title="E29">A61</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">j</font>
 ; <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> &amp; ( for <font color="Olive" title="b1">ai</font>, <font color="Olive" title="b2">aj</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 <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Olive" title="b1">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Olive" title="b2">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> holds <br/><font color="Olive" title="b1">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">aj</font> ) )</span><br/>

<a NAME="E3:14_2_1_5_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_5_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_5_1_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_2_1_5_1_1"/><b>then </b>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_5_1"><i><font color="Green" title="E29">A61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:14_2_1_5_1_1"/>
contradiction
 ; <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="E4:14_2_1_5_1"/><b>then </b><i><font color="Green" title="E30">A62</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">j</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_5_1"><i><font color="Green" title="E28">A60</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">j'</font> = <font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1;<br/>
<a NAME="E5:14_2_1_5_1"/>
( <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">l</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_5_1"><i><font color="Green" title="E30">A62</font></i></a>, <a class="txt" href="ramsey_1.html#E2:14_2_1_5_1"><i><font color="Green" title="E29">A61</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:14_2_1_5_1"/><b>then </b><i><font color="Green" title="E31">A63</font></i>: 
( <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">ai</font>, <font color="Olive" title="b2">aj'</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 <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 &amp; <font color="Olive" title="b1">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Olive" title="b2">aj'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">aj'</font> ) )
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_5"><i><font color="Green" title="E27">A59</font></i></a></i>;<br/>
<a NAME="E7:14_2_1_5_1"/>
<font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E14:14_2_1"><i><font color="Green" title="E24">A56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:14_2_1_5_1"/>
<font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E6:14_2_1_5_1"><i><font color="Green" title="E31">A63</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <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">ai</font>, <font color="Olive" title="b2">aj</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 <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Olive" title="b1">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Olive" title="b2">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> holds <br/><font color="Olive" title="b1">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">aj</font></span><br/>

<b>thus </b><a NAME="E9:14_2_1_5_1"/>
for <font color="Olive" title="b1">ai</font>, <font color="Olive" title="b2">aj</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 <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Olive" title="b1">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Olive" title="b2">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> holds <br/><font color="Olive" title="b1">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">aj</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">ai</font>, <font color="Maroon" title="c18">aj</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"> ( <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c17">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c18">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> implies <font color="Maroon" title="c17">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">aj</font> )</span><br/>

<b>assume </b><a NAME="E1:14_2_1_5_1_2"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</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="c17">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> or  not <font color="Maroon" title="c18">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> or <font color="Maroon" title="c17">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">aj</font> )</span><br/>

<b>assume </b><a NAME="E2:14_2_1_5_1_2"/><i><font color="Green" title="E32">A64</font></i>: 
( <font color="Maroon" title="c17">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c18">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">aj</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_5_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1_5_1_2_1"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 or <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_5_1_2_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_5_1_2_1_1"/><i><font color="Green" title="E33">A65</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">aj</font></span><br/><div class="add"><a NAME="E2:14_2_1_5_1_2_1_1"/>
<font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E14:14_2_1"><i><font color="Green" title="E24">A56</font></i></a></i>;<br/><b>hence </b><a NAME="E3:14_2_1_5_1_2_1_1"/>
<font color="Maroon" title="c17">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">aj</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_5_1_2"><i><font color="Green" title="E32">A64</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_5_1_2_1_1"><i><font color="Green" title="E33">A65</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</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="14_2_1_5_1_2_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_5_1_2_1_2"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">aj</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c19">aj'</font> = <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><a NAME="E3:14_2_1_5_1_2_1_2"/>
<font color="Maroon" title="c18">aj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">aj'</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_5_1_2"><i><font color="Green" title="E32">A64</font></i></a>, <a class="txt" href="ramsey_1.html#E14:14_2_1"><i><font color="Green" title="E24">A56</font></i></a></i>;<br/><a NAME="E4:14_2_1_5_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c19">aj'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">aj</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><b>hence </b><a NAME="E5:14_2_1_5_1_2_1_2"/>
<font color="Maroon" title="c17">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">aj</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E6:14_2_1_5_1"><i><font color="Green" title="E31">A63</font></i></a>, <a class="txt" href="ramsey_1.html#E2:14_2_1_5_1_2"><i><font color="Green" title="E32">A64</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>


</div><b>end;</b></div><a NAME="E17:14_2_1"/><i><font color="Green" title="E27">A66</font></i>: 
for <font color="Olive" title="b1">l</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">l</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="ramsey_1.html#E15:14_2_1"><i><font color="Green" title="E25">A57</font></i></a>, <a class="txt" href="ramsey_1.html#E16:14_2_1"><i><font color="Green" title="E26">A58</font></i></a>);<br/></i><a NAME="E18:14_2_1"/><i><font color="Green" title="E28">A67</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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 <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">j</font> holds <br/>( <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">j</font> &amp; ( for <font color="Olive" title="b3">ai</font>, <font color="Olive" title="b4">aj</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 <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b3">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Olive" title="b3">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b4">aj</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c14">j</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"> ( <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c14">j</font> implies ( <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">j</font> &amp; ( for <font color="Olive" title="b1">ai</font>, <font color="Olive" title="b2">aj</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 <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Olive" title="b1">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Olive" title="b2">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">j</font> holds <br/><font color="Olive" title="b1">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">aj</font> ) ) )</span><br/>

<b>assume </b><a NAME="E1:14_2_1_6"/><i><font color="Green" title="E29">A68</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c14">j</font>
 ; <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">j</font> &amp; ( for <font color="Olive" title="b1">ai</font>, <font color="Olive" title="b2">aj</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 <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Olive" title="b1">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Olive" title="b2">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">j</font> holds <br/><font color="Olive" title="b1">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">aj</font> ) )</span><br/>

<b>then consider </b><font color="Maroon" title="c15">l</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="E3:14_2_1_6"/><i><font color="Green" title="E30">A69</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">l</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<b>thus </b><a NAME="E4:14_2_1_6"/>
<font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E17:14_2_1"><i><font color="Green" title="E27">A66</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_6"><i><font color="Green" title="E29">A68</font></i></a>, <a class="txt" href="ramsey_1.html#E3:14_2_1_6"><i><font color="Green" title="E30">A69</font></i></a></i>; <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">ai</font>, <font color="Olive" title="b2">aj</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 <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Olive" title="b1">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Olive" title="b2">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">j</font> holds <br/><font color="Olive" title="b1">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">aj</font></span><br/>

<b>thus </b><a NAME="E5:14_2_1_6"/>
for <font color="Olive" title="b1">ai</font>, <font color="Olive" title="b2">aj</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 <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Olive" title="b1">ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Olive" title="b2">aj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">j</font> holds <br/><font color="Olive" title="b1">ai</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">aj</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E17:14_2_1"><i><font color="Green" title="E27">A66</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_6"><i><font color="Green" title="E29">A68</font></i></a>, <a class="txt" href="ramsey_1.html#E3:14_2_1_6"><i><font color="Green" title="E30">A69</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><a NAME="E19:14_2_1"/><i><font color="Green" title="E29">A70</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">Fi</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>  st <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">x</font> where <font color="Olive" title="b4">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x</font> &amp; <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">i</font> ) </span>}</span>  &amp; ( for <font color="Olive" title="b4">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b3">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) holds <br/>( <font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">Fi</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="c4">n</font>,<font color="Olive" title="b2">Y</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="14_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">Fi</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>  st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">x</font> where <font color="Olive" title="b3">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span>  &amp; ( for <font color="Olive" title="b3">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b2">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) holds <br/>( <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">Fi</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="c4">n</font>,<font color="Olive" title="b1">Y</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/><b>let </b><font color="Maroon" title="c14">Y</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"> for <font color="Olive" title="b1">Fi</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>  st <font color="Maroon" title="c14">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span>  &amp; ( for <font color="Olive" title="b2">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) holds <br/>( <font color="Maroon" title="c14">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/><b>let </b><font color="Maroon" title="c15">Fi</font> be   <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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</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">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span>  &amp; ( for <font color="Olive" title="b1">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c15">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) implies ( <font color="Maroon" title="c14">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> ) )</span><br/>





<b>assume </b><a NAME="E1:14_2_1_7"/><i><font color="Green" title="E30">A71</font></i>: 
<font color="Maroon" title="c14">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span> 
 ; <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">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st not <font color="Maroon" title="c15">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> or ( <font color="Maroon" title="c14">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> ) )</span><br/>

<b>assume </b><a NAME="E2:14_2_1_7"/><i><font color="Green" title="E31">A72</font></i>: 
for <font color="Olive" title="b1">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c15">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c16">x1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font>, <font color="Maroon" title="c17">y1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> <b> such that </b><br/><a NAME="E4:14_2_1_7"/><i><font color="Green" title="E32">A73</font></i>: 
( <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> &amp; <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c18">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">A</font>, <font color="Maroon" title="c19">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> <b> such that </b><br/><a NAME="E6:14_2_1_7"/><i><font color="Green" title="E33">A74</font></i>: 
( <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x2</font> &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y2</font> )
 ;<br/>
<a NAME="E7:14_2_1_7"/>
( <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x1</font> implies (  ex <font color="Olive" title="b1">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> ex <font color="Olive" title="b2">H1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st <br/>( not <font color="Olive" title="b2">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">F1</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="c4">n</font>,<font color="Olive" title="b2">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c19">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> &amp; <font color="Maroon" title="c17">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">y2</font> &amp; ( for <font color="Olive" title="b3">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b3">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">y1</font> &amp; <font color="Olive" title="b3">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H1</font> holds <br/><font color="Maroon" title="c19">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">y2'</font> ) ) &amp; ( not <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">x1</font> implies ( <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c19">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#E11:14_2_1"><i><font color="Green" title="E21">A49</font></i></a>, <a class="txt" href="ramsey_1.html#E4:14_2_1_7"><i><font color="Green" title="E32">A73</font></i></a>, <a class="txt" href="ramsey_1.html#E6:14_2_1_7"><i><font color="Green" title="E33">A74</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">F1</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>, <font color="Maroon" title="c21">H1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span><b> such that </b><br/><a NAME="E9:14_2_1_7"/><i><font color="Green" title="E34">A75</font></i>: 
( not <font color="Maroon" title="c21">H1</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c20">F1</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="c4">n</font>,<font color="Maroon" title="c21">H1</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c18">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">H1</font> &amp; ( for <font color="Olive" title="b1">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">x1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c20">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">y2'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>   st <font color="Olive" title="b1">y2'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">y1</font> &amp; <font color="Olive" title="b1">y2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">H1</font> holds <br/><font color="Maroon" title="c19">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y2'</font> ) )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_7"><i><font color="Green" title="E32">A73</font></i></a>, <a class="txt" href="ramsey_1.html#E14:14_2_1"><i><font color="Green" title="E24">A56</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">F1</font> = <font color="Maroon" title="c20">F1</font> 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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_7"><i><font color="Green" title="E32">A73</font></i></a></i>;<br/>
<a NAME="E11:14_2_1_7"/>
for <font color="Olive" title="b1">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c22">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x1'</font> be    <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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x1'</font></span><br/>

<b>thus </b><font color="Maroon" title="c22">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x1'</font> = 
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_7"><i><font color="Green" title="E32">A73</font></i></a>, <a class="txt" href="ramsey_1.html#E9:14_2_1_7"><i><font color="Green" title="E34">A75</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x1'</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_7"><i><font color="Green" title="E31">A72</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>
<a NAME="E12:14_2_1_7"/><b>then </b><i><font color="Green" title="E35">A76</font></i>: 
<font color="Maroon" title="c15">Fi</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="c4">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></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#E9:14_2_1_7"><i><font color="Green" title="E34">A75</font></i></a>, <a class="txt" href="ramsey_1.html#E6:14_2_1_7"><i><font color="Green" title="E33">A74</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_7_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">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="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font> implies <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> )</span><br/><b>assume </b><a NAME="E1:14_2_1_7_2"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">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="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c24">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> <b> such that </b><br/><a NAME="E3:14_2_1_7_2"/><i><font color="Green" title="E36">A77</font></i>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x'</font> &amp;  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x'</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_7"><i><font color="Green" title="E30">A71</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c25">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:14_2_1_7_2"/><i><font color="Green" title="E37">A78</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c25">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_7_2"><i><font color="Green" title="E36">A77</font></i></a></i>;<br/><a NAME="E6:14_2_1_7_2"/>
<font color="Maroon" title="c25">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="txt" href="ramsey_1.html#E5:14_2_1_7_2"><i><font color="Green" title="E37">A78</font></i></a></i>;<br/><a NAME="E7:14_2_1_7_2"/><b>then </b><i><font color="Green" title="E38">A79</font></i>: 
<font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E18:14_2_1"><i><font color="Green" title="E28">A67</font></i></a></i>;<br/><a NAME="E8:14_2_1_7_2"/>
<font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">j</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E14:14_2_1"><i><font color="Green" title="E24">A56</font></i></a></i>;<br/><b>hence </b><a NAME="E9:14_2_1_7_2"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E5:14_2_1_7_2"><i><font color="Green" title="E37">A78</font></i></a>, <a class="txt" href="ramsey_1.html#E7:14_2_1_7_2"><i><font color="Green" title="E38">A79</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>hence </b><a NAME="E14:14_2_1_7"/><i><font color="Green" title="E36">A80</font></i>: 
<font color="Maroon" title="c14">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a></span><br/>

<a NAME="E15:14_2_1_7"/><i><font color="Green" title="E37">A81</font></i>: 
 <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c14">Y</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="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E3"><i><font color="Green" title="E3">Lm1</font></i></a>, <a class="txt" href="ramsey_1.html#E14:14_2_1_7"><i><font color="Green" title="E36">A80</font></i></a></i>;<br/>
<a NAME="E16:14_2_1_7"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</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="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_7_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font>, <font color="Maroon" title="c24">y</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="c23">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="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c24">y</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="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> )</span><br/>

<b>assume </b><a NAME="E1:14_2_1_7_3"/>
<font color="Maroon" title="c23">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="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c24">y</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="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> )</span><br/>

<a NAME="E2:14_2_1_7_3"/><b>then </b><i><font color="Green" title="E38">A82</font></i>: 
( <font color="Maroon" title="c23">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> <font color="Maroon" title="c15">Fi</font> &amp; <font color="Maroon" title="c23">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="c4">n</font>,<font color="Maroon" title="c14">Y</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/>
<a NAME="E3:14_2_1_7_3"/><b>then </b><i><font color="Green" title="E39">A83</font></i>: 
<font color="Maroon" title="c23">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="c15">Fi</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="c4">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E15:14_2_1_7"><i><font color="Green" title="E37">A81</font></i></a>, <a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/>
<b>assume </b><a NAME="E4:14_2_1_7_3"/>
<font color="Maroon" title="c24">y</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="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font></span><br/>

<a NAME="E5:14_2_1_7_3"/><b>then </b><i><font color="Green" title="E40">A84</font></i>: 
( <font color="Maroon" title="c24">y</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> <font color="Maroon" title="c15">Fi</font> &amp; <font color="Maroon" title="c24">y</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="c4">n</font>,<font color="Maroon" title="c14">Y</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/>
<a NAME="E6:14_2_1_7_3"/><b>then </b><i><font color="Green" title="E41">A85</font></i>: 
<font color="Maroon" title="c24">y</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="c15">Fi</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="c4">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E15:14_2_1_7"><i><font color="Green" title="E37">A81</font></i></a>, <a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Fi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c4">n</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E15:14_2_1_7"><i><font color="Green" title="E37">A81</font></i></a>, <a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">Fi</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="c4">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_7_3"><i><font color="Green" title="E38">A82</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="c15">Fi</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="c4">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E6:14_2_1_7_3"><i><font color="Green" title="E41">A85</font></i></a>, <a class="txt" href="ramsey_1.html#E12:14_2_1_7"><i><font color="Green" title="E35">A76</font></i></a>, <a class="txt" href="ramsey_1.html#E3:14_2_1_7_3"><i><font color="Green" title="E39">A83</font></i></a>, <a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">Fi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c4">n</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E5:14_2_1_7_3"><i><font color="Green" title="E40">A84</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="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E15:14_2_1_7"><i><font color="Green" title="E37">A81</font></i></a>, <a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</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="E17:14_2_1_7"/>
<font color="Maroon" title="c15">Fi</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="c4">n</font>,<font color="Maroon" title="c14">Y</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</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="E20:14_2_1"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</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> <font color="Maroon" title="c12">a</font> &amp; <font color="Olive" title="b2">x2</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> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c14">x2</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="c13">x1</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> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c14">x2</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> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font> implies <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x2</font> )</span><br/>

<b>assume </b><a NAME="E1:14_2_1_8"/><i><font color="Green" title="E30">A86</font></i>: 
<font color="Maroon" title="c13">x1</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> <font color="Maroon" title="c12">a</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="c14">x2</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> <font color="Maroon" title="c12">a</font> or  not <font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font> or <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x2</font> )</span><br/>

<b>assume </b><a NAME="E2:14_2_1_8"/><i><font color="Green" title="E31">A87</font></i>: 
<font color="Maroon" title="c14">x2</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> <font color="Maroon" title="c12">a</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="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font> or <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x2</font> )</span><br/>

<b>assume </b><a NAME="E3:14_2_1_8"/><i><font color="Green" title="E32">A88</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x2</font></span><br/>

<b>assume </b><a NAME="E4:14_2_1_8"/><i><font color="Green" title="E33">A89</font></i>: 
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>reconsider </b><font color="Maroon" title="c15">i1</font> = <font color="Maroon" title="c13">x1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_8"><i><font color="Green" title="E30">A86</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">i2</font> = <font color="Maroon" title="c14">x2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_8"><i><font color="Green" title="E31">A87</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">ai1</font> = <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c18">ai2</font> = <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1_8_1"/>
( <font color="Maroon" title="c15">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">i2</font> or <font color="Maroon" title="c15">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">i2</font> )
 </span><b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_8"><i><font color="Green" title="E33">A89</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_8_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_8_1_1"/>
<font color="Maroon" title="c15">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:14_2_1_8_1_1"/><b>then </b>
<font color="Maroon" title="c17">ai1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">ai2</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E18:14_2_1"><i><font color="Green" title="E28">A67</font></i></a></i>;<br/><b>hence </b><a NAME="E3:14_2_1_8_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_8"><i><font color="Green" title="E32">A88</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_8_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_8_1_2"/>
<font color="Maroon" title="c15">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:14_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c17">ai1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">ai2</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E18:14_2_1"><i><font color="Green" title="E28">A67</font></i></a></i>;<br/><b>hence </b><a NAME="E3:14_2_1_8_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_8"><i><font color="Green" title="E32">A88</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></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E21:14_2_1"/><b>then </b><i><font color="Green" title="E30">A90</font></i>: 
<font color="Maroon" title="c12">a</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E22:14_2_1"/><i><font color="Green" title="E31">A91</font></i>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">a</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="E23:14_2_1"/><i><font color="Green" title="E32">A92</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x'</font> where <font color="Olive" title="b2">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x'</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">i</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><br/>

<b>set </b><font color="Maroon" title="c14">Z</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span> ;<br/>
<i><font color="Green" title="E33">A93</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></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 href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></i>
<br/>.= 
<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>

;<br/>
<a NAME="E2:14_2_1_9"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x'</font> where <font color="Olive" title="b2">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x'</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span>  iff <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">z</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="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span>  iff <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> )</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="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> implies <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span>  )</span>
<div class="add"><b>assume </b><a NAME="E1:14_2_1_9_2_1"/>
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c16">z'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> <b> such that </b><br/><a NAME="E3:14_2_1_9_2_1"/><i><font color="Green" title="E34">A94</font></i>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">z'</font> &amp;  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">z'</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c17">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:14_2_1_9_2_1"/><i><font color="Green" title="E35">A95</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_9_2_1"><i><font color="Green" title="E34">A94</font></i></a></i>;<br/><a NAME="E6:14_2_1_9_2_1"/>
<font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="txt" href="ramsey_1.html#E5:14_2_1_9_2_1"><i><font color="Green" title="E35">A95</font></i></a></i>;<br/><a NAME="E7:14_2_1_9_2_1"/><b>then </b>
not <font color="Maroon" title="c17">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/><a NAME="E8:14_2_1_9_2_1"/><b>then </b>
( <font color="Maroon" title="c17">j</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> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c17">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E22:14_2_1"><i><font color="Green" title="E31">A91</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E9:14_2_1_9_2_1"/>
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E5:14_2_1_9_2_1"><i><font color="Green" title="E35">A95</font></i></a>, <a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</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="E2:14_2_1_9_2"/>
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c16">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:14_2_1_9_2"/><i><font color="Green" title="E34">A96</font></i>: 
( <font color="Maroon" title="c16">j</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="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">j</font> = <font color="Maroon" title="c16">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_9_2"><i><font color="Green" title="E34">A96</font></i></a></i>;<br/>
<a NAME="E6:14_2_1_9_2"/>
not <font color="Maroon" title="c17">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_9_2"><i><font color="Green" title="E34">A96</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_9"><i><font color="Green" title="E33">A93</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:14_2_1_9_2"/><b>then </b>
<font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/>
<a NAME="E8:14_2_1_9_2"/><b>then </b>
( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c17">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> )
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_9_2"><i><font color="Green" title="E34">A96</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E9:14_2_1_9_2"/>
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span> 
 ; <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="E3:14_2_1_9"/><b>then </b><i><font color="Green" title="E34">A97</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E4:14_2_1_9"/><i><font color="Green" title="E35">A98</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E21:14_2_1"><i><font color="Green" title="E30">A90</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_9_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">z</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="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span>  implies <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/><b>assume </b><a NAME="E1:14_2_1_9_3"/>
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><b>then consider </b><font color="Maroon" title="c16">z'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> <b> such that </b><br/><a NAME="E3:14_2_1_9_3"/><i><font color="Green" title="E36">A99</font></i>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">z'</font> &amp;  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">z'</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) )
 ;<br/><b>thus </b><a NAME="E4:14_2_1_9_3"/>
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_9_3"><i><font color="Green" title="E36">A99</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>
<a NAME="E6:14_2_1_9"/><b>then </b><i><font color="Green" title="E36">A100</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:14_2_1_9"/>
not  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> 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#E1:14_2_1_9"><i><font color="Green" title="E33">A93</font></i></a>, <a class="ref" href="ramsey_1.html#T4" target="_self" title="RAMSEY_1:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E8:14_2_1_9"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">i</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_9"><i><font color="Green" title="E34">A97</font></i></a>, <a class="txt" href="ramsey_1.html#E6:14_2_1_9"><i><font color="Green" title="E36">A100</font></i></a>, <a class="ref" href="ramsey_1.html#T2" target="_self" title="RAMSEY_1:th.2">Th2</a>, <a class="txt" href="ramsey_1.html#E4:14_2_1_9"><i><font color="Green" title="E35">A98</font></i></a>, <a class="ref" href="card_1.html#T97" title="CARD_1:th.97">CARD_1:97</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>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">Fi</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">x</font> where <font color="Olive" title="b4">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x</font> &amp; <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">i</font> ) </span>}</span>  &amp; ( for <font color="Olive" title="b4">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b3">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) holds <br/> ex <font color="Olive" title="b4">l</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/>( <font color="Olive" title="b4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b4">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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="b3">Fi</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="c4">n</font>,<font color="Olive" title="b1">Y</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"><font color="Olive" title="b4">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> );<br/><a NAME="E24:14_2_1"/><i><font color="Green" title="E33">A101</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c13">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:14_2_1_10"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c13">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">i'</font> = <font color="Maroon" title="c13">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>set </b><font color="Maroon" title="c15">Y'</font> = <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">a'</font> = <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E14:14_2_1"><i><font color="Green" title="E24">A56</font></i></a></i>;<br/>
<a NAME="E4:14_2_1_10"/>
<font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c17">Y''</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c6">X</font><b> such that </b><br/><a NAME="E6:14_2_1_10"/><i><font color="Green" title="E34">A102</font></i>: 
( <font color="Maroon" title="c17">Y''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c17">Y''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 ;<br/>
<b>consider </b><font color="Maroon" title="c18">Fa</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>, <font color="Maroon" title="c19">Ha</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span><b> such that </b><br/><a NAME="E8:14_2_1_10"/><i><font color="Green" title="E35">A103</font></i>: 
( not <font color="Maroon" title="c19">Ha</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c18">Fa</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="c4">n</font>,<font color="Maroon" title="c19">Ha</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; ( for <font color="Olive" title="b1">Y''</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c18">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y''</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E6:14_2_1_10"><i><font color="Green" title="E34">A102</font></i></a>, <a class="txt" href="ramsey_1.html#E4:14_2_1"><i><font color="Green" title="E18">A11</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">Z</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span> ;<br/>
<a NAME="E9:14_2_1_10"/><i><font color="Green" title="E36">A104</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c18">Fa</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="c4">n</font>,<span class="p2">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span> </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#E19:14_2_1"><i><font color="Green" title="E29">A70</font></i></a>, <a class="txt" href="ramsey_1.html#E8:14_2_1_10"><i><font color="Green" title="E35">A103</font></i></a></i>;<br/>
<a NAME="E10:14_2_1_10"/><i><font color="Green" title="E37">A105</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font> &amp; <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font> &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; not <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E14:14_2_1"><i><font color="Green" title="E24">A56</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">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="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span>  implies <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:14_2_1_10_1"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E2:14_2_1_10_1"/><b>then </b><i><font color="Green" title="E38">A106</font></i>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E9:14_2_1_10"><i><font color="Green" title="E36">A104</font></i></a></i>;<br/><a NAME="E3:14_2_1_10_1"/>
not <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_10_1"><i><font color="Green" title="E38">A106</font></i></a>, <a class="txt" href="ramsey_1.html#E10:14_2_1_10"><i><font color="Green" title="E37">A105</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:14_2_1_10_1"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_10_1"><i><font color="Green" title="E38">A106</font></i></a>, <a class="txt" href="ramsey_1.html#E10:14_2_1_10"><i><font color="Green" title="E37">A105</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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="E12:14_2_1_10"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E13:14_2_1_10"/><b>then </b><i><font color="Green" title="E38">A107</font></i>: 
 <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c4">n</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span>  <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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<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="E14:14_2_1_10"/><i><font color="Green" title="E39">A108</font></i>: 
<font color="Maroon" title="c18">Fa</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="c4">n</font>,<span class="p2">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span> </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> <font color="Maroon" title="c4">n</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span> ,<font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="ramsey_1.html#E13:14_2_1_10"><i><font color="Green" title="E38">A107</font></i></a>, <a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></i>;<br/>
<a NAME="E15:14_2_1_10"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E23:14_2_1"><i><font color="Green" title="E32">A92</font></i></a></i>;<br/>
<a NAME="E16:14_2_1_10"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">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> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span>  &amp; not <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="ref" href="card_4.html#T13" title="CARD_4:th.13">CARD_4:13</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<a NAME="E17:14_2_1_10"/><b>then </b>
not  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c4">n</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="group_10.html#T2" title="GROUP_10:th.2">GROUP_10:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">k</font><b> such that </b><br/><a NAME="E19:14_2_1_10"/><i><font color="Green" title="E40">A109</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fa</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="c4">n</font>,<span class="p3">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i'</font> ) </span>}</span> </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"><font color="Maroon" title="c21">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="ramsey_1.html#E14:14_2_1_10"><i><font color="Green" title="E39">A108</font></i></a>, <a class="txt" href="ramsey_1.html#E9:14_2_1_10"><i><font color="Green" title="E36">A104</font></i></a>, <a class="txt" href="ramsey_1.html#E10"><i><font color="Green" title="E10">Lm4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">y</font> = <font color="Maroon" title="c21">y</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c22">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="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c22">y</font>] )</span><br/>

<b>thus </b><a NAME="E21:14_2_1_10"/><i><font color="Green" title="E41">A110</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1"><i><font color="Green" title="E17">A10</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">S<sub>5</sub></font>[<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c22">y</font>]</span><br/>

<b>thus </b><a NAME="E22:14_2_1_10"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c22">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_10_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:14_2_1_10_2"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">Fi</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">x'</font> where <font color="Olive" title="b4">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x'</font> &amp; <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">i</font> ) </span>}</span>  &amp; ( for <font color="Olive" title="b4">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b3">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) holds <br/> ex <font color="Olive" title="b4">l</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/>( <font color="Olive" title="b4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Olive" title="b4">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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="b3">Fi</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="c4">n</font>,<font color="Olive" title="b1">Y</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"><font color="Olive" title="b4">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_10_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">Y</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"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">Fi</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c23">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">x'</font> where <font color="Olive" title="b3">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x'</font> &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">i</font> ) </span>}</span>  &amp; ( for <font color="Olive" title="b3">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b2">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) holds <br/> ex <font color="Olive" title="b3">l</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/>( <font color="Olive" title="b3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Olive" title="b3">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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">Fi</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="c4">n</font>,<font color="Maroon" title="c23">Y</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"><font color="Olive" title="b3">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><b>let </b><font color="Maroon" title="c24">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">Fi</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>  st <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c23">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x'</font> where <font color="Olive" title="b2">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x'</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">i</font> ) </span>}</span>  &amp; ( for <font color="Olive" title="b2">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Olive" title="b1">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) holds <br/> ex <font color="Olive" title="b2">l</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/>( <font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Olive" title="b2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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="b1">Fi</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="c4">n</font>,<font color="Maroon" title="c23">Y</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"><font color="Olive" title="b2">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><b>let </b><font color="Maroon" title="c25">Fi</font> be   <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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c23">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">i</font> ) </span>}</span>  &amp; ( for <font color="Olive" title="b1">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c25">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) implies  ex <font color="Olive" title="b1">l</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/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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="c25">Fi</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="c4">n</font>,<font color="Maroon" title="c23">Y</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"><font color="Olive" title="b1">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )</span><br/>





<b>assume </b><a NAME="E1:14_2_1_10_2_1"/><i><font color="Green" title="E42">A111</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</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="c23">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">i</font> ) </span>}</span>  or  ex <font color="Olive" title="b1">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st not <font color="Maroon" title="c25">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> or  ex <font color="Olive" title="b1">l</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/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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="c25">Fi</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="c4">n</font>,<font color="Maroon" title="c23">Y</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"><font color="Olive" title="b1">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )</span><br/>

<b>assume </b><a NAME="E2:14_2_1_10_2_1"/><i><font color="Green" title="E43">A112</font></i>: 
<font color="Maroon" title="c23">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x'</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">i</font> ) </span>}</span> 
 ; <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">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> st not <font color="Maroon" title="c25">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> or  ex <font color="Olive" title="b1">l</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/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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="c25">Fi</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="c4">n</font>,<font color="Maroon" title="c23">Y</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"><font color="Olive" title="b1">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )</span><br/>

<b>assume </b><a NAME="E3:14_2_1_10_2_1"/><i><font color="Green" title="E44">A113</font></i>: 
for <font color="Olive" title="b1">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c25">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ; <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">l</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/>( <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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="c25">Fi</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="c4">n</font>,<font color="Maroon" title="c23">Y</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"><font color="Olive" title="b1">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c26">k'</font> = <font color="Maroon" title="c5">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:14_2_1_10_2_1"/>
<font color="Maroon" title="c26">k'</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T8" title="STIRL2_1:th.8">STIRL2_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c27">l</font> = <font color="Maroon" title="c22">y</font> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="ramsey_1.html#E21:14_2_1_10"><i><font color="Green" title="E41">A110</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c27">l</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c27">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Maroon" title="c27">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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="c25">Fi</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="c4">n</font>,<font color="Maroon" title="c23">Y</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"><font color="Maroon" title="c27">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<b>thus </b><a NAME="E7:14_2_1_10_2_1"/>
<font color="Maroon" title="c27">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">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="c27">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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="c25">Fi</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="c4">n</font>,<font color="Maroon" title="c23">Y</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"><font color="Maroon" title="c27">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<b>thus </b><a NAME="E8:14_2_1_10_2_1"/>
<font color="Maroon" title="c27">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1"><i><font color="Green" title="E17">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">Fi</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="c4">n</font>,<font color="Maroon" title="c23">Y</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"><font color="Maroon" title="c27">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E9:14_2_1_10_2_1"/>
for <font color="Olive" title="b1">x1'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c18">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_10_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c28">x1'</font> be    <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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x1'</font></span><br/>

<b>thus </b><font color="Maroon" title="c18">Fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x1'</font> = 
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">x1'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="ramsey_1.html#E1:14_2_1_10_2_1"><i><font color="Green" title="E42">A111</font></i></a>, <a class="txt" href="ramsey_1.html#E8:14_2_1_10"><i><font color="Green" title="E35">A103</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c25">Fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x1'</font>
<b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_10_2_1"><i><font color="Green" title="E44">A113</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>hence </b><a NAME="E10:14_2_1_10_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">Fi</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="c4">n</font>,<font color="Maroon" title="c23">Y</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"><font color="Maroon" title="c27">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E19:14_2_1_10"><i><font color="Green" title="E40">A109</font></i></a>, <a class="txt" href="ramsey_1.html#E2:14_2_1_10_2_1"><i><font color="Green" title="E43">A112</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_10_2_1"><i><font color="Green" title="E42">A111</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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><b>end;</b></div><b>consider </b><font color="Maroon" title="c13">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c5">k</font><b> such that </b><br/><a NAME="E26:14_2_1"/><i><font color="Green" title="E34">A114</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="ramsey_1.html#E24:14_2_1"><i><font color="Green" title="E33">A101</font></i></a>);<br/></i><a NAME="E27:14_2_1"/><i><font color="Green" title="E35">A115</font></i>: 
not  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">g</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#E3:14_2_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E28:14_2_1"/>
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">g'</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E30:14_2_1"/><i><font color="Green" title="E36">A116</font></i>: 
( <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g'</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">k</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E31:14_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</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#E30:14_2_1"><i><font color="Green" title="E36">A116</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">k'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E33:14_2_1"/><i><font color="Green" title="E37">A117</font></i>: 
( <font color="Maroon" title="c15">k'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> &amp; not <font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> 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#E27:14_2_1"><i><font color="Green" title="E35">A115</font></i></a>, <a class="ref" href="compl_sp.html#T24" title="COMPL_SP:th.24">COMPL_SP:24</a></i>;<br/><b>set </b><font color="Maroon" title="c16">H</font> = <font color="Maroon" title="c12">a</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/><a NAME="E34:14_2_1"/><i><font color="Green" title="E38">A118</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T167" title="RELAT_1:th.167">RELAT_1:167</a></i>;<br/><a NAME="E35:14_2_1"/>
<font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c12">a</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="ramsey_1.html#E21:14_2_1"><i><font color="Green" title="E30">A90</font></i></a>, <a class="txt" href="ramsey_1.html#E22:14_2_1"><i><font color="Green" title="E31">A91</font></i></a>, <a class="ref" href="card_1.html#T60" title="CARD_1:th.60">CARD_1:60</a>, <a class="txt" href="ramsey_1.html#E34:14_2_1"><i><font color="Green" title="E38">A118</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E36:14_2_1"/><b>then </b><i><font color="Green" title="E39">A119</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <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"><font color="Maroon" title="c12">a</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/><a NAME="E37:14_2_1"/><i><font color="Green" title="E40">A120</font></i>: 
not  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> 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#E33:14_2_1"><i><font color="Green" title="E37">A117</font></i></a>, <a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">y</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="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> implies <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> )</span><br/><b>assume </b><a NAME="E1:14_2_1_11"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font></span><br/><b>then consider </b><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_2_1_11"/><i><font color="Green" title="E41">A121</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></i>;<br/><a NAME="E4:14_2_1_11"/><i><font color="Green" title="E42">A122</font></i>: 
( <font color="Maroon" title="c18">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> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_11"><i><font color="Green" title="E41">A121</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">i</font> = <font color="Maroon" title="c18">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E6:14_2_1_11"/><i><font color="Green" title="E43">A123</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_11"><i><font color="Green" title="E42">A122</font></i></a>, <a class="txt" href="ramsey_1.html#E14:14_2_1"><i><font color="Green" title="E24">A56</font></i></a></i>;<br/><a NAME="E7:14_2_1_11"/>
<font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c6">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c20">Xi</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c6">X</font><b> such that </b><br/><a NAME="E9:14_2_1_11"/><i><font color="Green" title="E44">A124</font></i>: 
( <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Xi</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c20">Xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 ;<br/><b>thus </b><a NAME="E10:14_2_1_11"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E9:14_2_1_11"><i><font color="Green" title="E44">A124</font></i></a>, <a class="txt" href="ramsey_1.html#E6:14_2_1_11"><i><font color="Green" title="E43">A123</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>then reconsider </b><font color="Maroon" title="c17">H</font> = <font color="Maroon" title="c12">a</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c6">X</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>take </b><font color="Maroon" title="c18">H</font> = <font color="Maroon" title="c17">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="c18">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c7">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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )</span><br/><b>thus </b><a NAME="E40:14_2_1"/>
not <font color="Maroon" title="c18">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#E37:14_2_1"><i><font color="Green" title="E40">A120</font></i></a>, <a class="txt" href="ramsey_1.html#E36:14_2_1"><i><font color="Green" title="E39">A119</font></i></a>, <a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</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="c7">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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a></span><br/><a NAME="E41:14_2_1"/><i><font color="Green" title="E41">A125</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</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="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">y</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="c19">y</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="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k'</font> )</span><br/>

<b>assume </b><a NAME="E1:14_2_1_12"/>
<font color="Maroon" title="c19">y</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="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k'</font></span><br/>

<a NAME="E2:14_2_1_12"/><b>then </b><i><font color="Green" title="E42">A126</font></i>: 
( <font color="Maroon" title="c19">y</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> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c19">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">Y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c18">H</font><b> such that </b><br/><a NAME="E4:14_2_1_12"/><i><font color="Green" title="E43">A127</font></i>: 
( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Y</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c20">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 )
 ;<br/>
<b>set </b><font color="Maroon" title="c21">y0</font> =  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font>;<br/>
<a NAME="E5:14_2_1_12"/>
<font color="Maroon" title="c20">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:14_2_1_12"/><b>then </b><i><font color="Green" title="E44">A128</font></i>: 
<font color="Maroon" title="c20">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:14_2_1_12"/><b>then </b><i><font color="Green" title="E45">A129</font></i>: 
(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Y</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Y</font> holds <br/> <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_12"><i><font color="Green" title="E43">A127</font></i></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:14_2_1_12"/><i><font color="Green" title="E46">A130</font></i>: 
( <font color="Maroon" title="c22">x0</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> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c22">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x0</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">y0'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:14_2_1_12"/><i><font color="Green" title="E47">A131</font></i>: 
( <font color="Maroon" title="c23">y0'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">x0</font>,<font color="Maroon" title="c23">y0'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> &amp; <font color="Maroon" title="c23">y0'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E9:14_2_1_12"><i><font color="Green" title="E46">A130</font></i></a>, <a class="ref" href="relat_1.html#T166" title="RELAT_1:th.166">RELAT_1:166</a></i>;<br/>
<a NAME="E12:14_2_1_12"/><i><font color="Green" title="E48">A132</font></i>: 
( <font color="Maroon" title="c22">x0</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> <font color="Maroon" title="c13">g</font> &amp; <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y0'</font> )
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E11:14_2_1_12"><i><font color="Green" title="E47">A131</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">x0</font> = <font color="Maroon" title="c22">x0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="ramsey_1.html#E12:14_2_1_12"><i><font color="Green" title="E48">A132</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">a'</font> = <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font> <b>by </b><i><a class="txt" href="ramsey_1.html#E14:14_2_1"><i><font color="Green" title="E24">A56</font></i></a></i>;<br/>
<a NAME="E15:14_2_1_12"/>
<font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</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> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<font color="Maroon" title="c6">X</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c26">S0</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c6">X</font><b> such that </b><br/><a NAME="E17:14_2_1_12"/><i><font color="Green" title="E49">A133</font></i>: 
( <font color="Maroon" title="c26">S0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c26">S0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 ;<br/>
<b>consider </b><font color="Maroon" title="c27">F0</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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c25">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c5">k</font>, <font color="Maroon" title="c28">H0</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c25">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span><b> such that </b><br/><a NAME="E19:14_2_1_12"/><i><font color="Green" title="E50">A134</font></i>: 
( not <font color="Maroon" title="c28">H0</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c27">F0</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="c4">n</font>,<font color="Maroon" title="c28">H0</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; ( for <font color="Olive" title="b1">Y'</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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c25">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds  <font color="Maroon" title="c27">F0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c25">a'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="ramsey_1.html#E17:14_2_1_12"><i><font color="Green" title="E49">A133</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c29">Y0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c30">l</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="E21:14_2_1_12"/><i><font color="Green" title="E51">A135</font></i>: 
( <font color="Maroon" title="c30">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font> &amp; <font color="Maroon" title="c30">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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="c27">F0</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="c4">n</font>,<span class="p3">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> </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"><font color="Maroon" title="c30">l</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E26:14_2_1"><i><font color="Green" title="E34">A114</font></i></a>, <a class="txt" href="ramsey_1.html#E19:14_2_1_12"><i><font color="Green" title="E50">A134</font></i></a></i>;<br/>
<a NAME="E22:14_2_1_12"/><i><font color="Green" title="E52">A136</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">F0</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="c4">n</font>,<span class="p3">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> </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"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E21:14_2_1_12"><i><font color="Green" title="E51">A135</font></i></a>, <a class="txt" href="ramsey_1.html#E12:14_2_1_12"><i><font color="Green" title="E48">A132</font></i></a>, <a class="txt" href="ramsey_1.html#E11:14_2_1_12"><i><font color="Green" title="E47">A131</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c31">Y'</font> = <font color="Maroon" title="c19">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E23:14_2_1_12"/><i><font color="Green" title="E53">A137</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c27">F0</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="c4">n</font>,<span class="p2">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> </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#E19:14_2_1"><i><font color="Green" title="E29">A70</font></i></a>, <a class="txt" href="ramsey_1.html#E19:14_2_1_12"><i><font color="Green" title="E50">A134</font></i></a></i>;<br/>
<a NAME="E24:14_2_1_12"/><i><font color="Green" title="E54">A138</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font> &amp; <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font> &amp; <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> &amp; not <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E14:14_2_1"><i><font color="Green" title="E24">A56</font></i></a></i>;<br/>
<a NAME="E25:14_2_1_12"/><i><font color="Green" title="E55">A139</font></i>: 
not <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E23:14_2_1_12"><i><font color="Green" title="E53">A137</font></i></a>, <a class="txt" href="ramsey_1.html#E14:14_2_1"><i><font color="Green" title="E24">A56</font></i></a></i>;<br/>
<a NAME="E26:14_2_1_12"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E23:14_2_1_12"><i><font color="Green" title="E53">A137</font></i></a>, <a class="txt" href="ramsey_1.html#E24:14_2_1_12"><i><font color="Green" title="E54">A138</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E27:14_2_1_12"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E28:14_2_1_12"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E25:14_2_1_12"><i><font color="Green" title="E55">A139</font></i></a>, <a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>;<br/>
<a NAME="E29:14_2_1_12"/><b>then </b><i><font color="Green" title="E56">A140</font></i>: 
 <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c4">n</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span>  <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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E3"><i><font color="Green" title="E3">Lm1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c32">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="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span>  )</span><br/><b>assume </b><a NAME="E1:14_2_1_12_1"/><i><font color="Green" title="E57">A141</font></i>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> </span><br/><a NAME="E2:14_2_1_12_1"/><b>then </b><i><font color="Green" title="E58">A142</font></i>: 
( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">y</font> &amp; not <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c33">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:14_2_1_12_1"/><i><font color="Green" title="E59">A143</font></i>: 
( <font color="Maroon" title="c33">j</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> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c33">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k'</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">j</font> )
 <b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_12_1"><i><font color="Green" title="E58">A142</font></i></a>, <a class="txt" href="ramsey_1.html#E4:14_2_1_12"><i><font color="Green" title="E43">A127</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c34">x'</font> = <font color="Maroon" title="c32">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_12_1"><i><font color="Green" title="E58">A142</font></i></a>, <a class="txt" href="ramsey_1.html#E4:14_2_1_12"><i><font color="Green" title="E43">A127</font></i></a>, <a class="txt" href="ramsey_1.html#E6:14_2_1_12"><i><font color="Green" title="E44">A128</font></i></a></i>;<br/><a NAME="E6:14_2_1_12_1"/>
 ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x'</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_12_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c35">j</font> = <font color="Maroon" title="c33">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_12_1"><i><font color="Green" title="E59">A143</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c35">j</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x'</font> &amp; <font color="Maroon" title="c35">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> )</span><br/>

<b>thus </b><a NAME="E2:14_2_1_12_1_1"/>
<font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x'</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_12_1"><i><font color="Green" title="E59">A143</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="c35">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font></span><br/>

<a NAME="E3:14_2_1_12_1_1"/><i><font color="Green" title="E60">A144</font></i>: 
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">x'</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E6:14_2_1_12"><i><font color="Green" title="E44">A128</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_12_1"><i><font color="Green" title="E57">A141</font></i></a>, <a class="txt" href="ramsey_1.html#E4:14_2_1_12"><i><font color="Green" title="E43">A127</font></i></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></i>;<br/>
<b>thus </b><a NAME="E4:14_2_1_12_1_1"/>
<font color="Maroon" title="c35">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_12_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_12_1_1_1"/><i><font color="Green" title="E61">A145</font></i>: 
<font color="Maroon" title="c24">x0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:14_2_1_12_1_1_1"/>
<font color="Maroon" title="c24">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c35">j</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_12_1"><i><font color="Green" title="E59">A143</font></i></a>, <a class="txt" href="ramsey_1.html#E2:14_2_1_12_1"><i><font color="Green" title="E58">A142</font></i></a>, <a class="txt" href="ramsey_1.html#E9:14_2_1_12"><i><font color="Green" title="E46">A130</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:14_2_1_12_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1_12_1_1"><i><font color="Green" title="E60">A144</font></i></a>, <a class="txt" href="ramsey_1.html#E9:14_2_1_12"><i><font color="Green" title="E46">A130</font></i></a>, <a class="txt" href="ramsey_1.html#E1:14_2_1_12_1_1_1"><i><font color="Green" title="E61">A145</font></i></a>, <a class="txt" href="ramsey_1.html#E4:14_2_1_12_1"><i><font color="Green" title="E59">A143</font></i></a>, <a class="txt" href="ramsey_1.html#E18:14_2_1"><i><font color="Green" title="E28">A67</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>


</div><b>end;</b></div><b>hence </b><a NAME="E7:14_2_1_12_1"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> 
 ; <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="E31:14_2_1_12"/><b>then </b><i><font color="Green" title="E57">A146</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E32:14_2_1_12"/><i><font color="Green" title="E58">A147</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">y</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_12"><i><font color="Green" title="E43">A127</font></i></a>, <a class="txt" href="ramsey_1.html#E7:14_2_1_12"><i><font color="Green" title="E45">A129</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E33:14_2_1_12"/><i><font color="Green" title="E59">A148</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E9:14_2_1_12"><i><font color="Green" title="E46">A130</font></i></a>, <a class="txt" href="ramsey_1.html#E32:14_2_1_12"><i><font color="Green" title="E58">A147</font></i></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">y'</font> = <font color="Maroon" title="c19">y</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_12"><i><font color="Green" title="E43">A127</font></i></a>, <a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a></i>;<br/>
<a NAME="E35:14_2_1_12"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c32">y'</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c33">Y''</font> = <font color="Maroon" title="c19">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="card_2.html#T68" title="CARD_2:th.68">CARD_2:68</a></i>;<br/>
<a NAME="E37:14_2_1_12"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></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>;<br/>
<a NAME="E38:14_2_1_12"/><b>then </b><i><font color="Green" title="E60">A149</font></i>: 
not <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E9:14_2_1_12"><i><font color="Green" title="E46">A130</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E39:14_2_1_12"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c32">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c33">Y''</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E33:14_2_1_12"><i><font color="Green" title="E59">A148</font></i></a>, <a class="txt" href="ramsey_1.html#E38:14_2_1_12"><i><font color="Green" title="E60">A149</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/>
<a NAME="E40:14_2_1_12"/><b>then </b><i><font color="Green" title="E61">A150</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="c4">n</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E4:14_2_1_12"><i><font color="Green" title="E43">A127</font></i></a>, <a class="txt" href="ramsey_1.html#E31:14_2_1_12"><i><font color="Green" title="E57">A146</font></i></a></i>;<br/>
<a NAME="E41:14_2_1_12"/><b>then </b><i><font color="Green" title="E62">A151</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E29:14_2_1_12"><i><font color="Green" title="E56">A140</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">Y'</font> = <font color="Maroon" title="c19">y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c20">Y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> 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> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="ramsey_1.html#E29:14_2_1_12"><i><font color="Green" title="E56">A140</font></i></a>, <a class="txt" href="ramsey_1.html#E40:14_2_1_12"><i><font color="Green" title="E61">A150</font></i></a></i>;<br/>
<a NAME="E43:14_2_1_12"/>
<font color="Maroon" title="c34">Y'</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> <font color="Maroon" title="c27">F0</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E3:14_2_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="txt" href="ramsey_1.html#E41:14_2_1_12"><i><font color="Green" title="E62">A151</font></i></a></i>;<br/>
<a NAME="E44:14_2_1_12"/><b>then </b>
<font color="Maroon" title="c34">Y'</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="c27">F0</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="c4">n</font>,<span class="p3">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E40:14_2_1_12"><i><font color="Green" title="E61">A150</font></i></a>, <a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/>
<a NAME="E45:14_2_1_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">F0</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="c4">n</font>,<span class="p3">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">Y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">F0</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="c4">n</font>,<span class="p3">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E46:14_2_1_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">F0</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="c4">n</font>,<span class="p3">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  :  ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">x0</font> ) </span>}</span> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k'</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E22:14_2_1_12"><i><font color="Green" title="E52">A136</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E47:14_2_1_12"/><b>then </b><i><font color="Green" title="E63">A152</font></i>: 
<font color="Maroon" title="c27">F0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k'</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E40:14_2_1_12"><i><font color="Green" title="E61">A150</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E48:14_2_1_12"/>
<font color="Maroon" title="c27">F0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">Y'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E19:14_2_1_12"><i><font color="Green" title="E50">A134</font></i></a></i>;<br/>
<b>hence </b><a NAME="E49:14_2_1_12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k'</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E47:14_2_1_12"><i><font color="Green" title="E63">A152</font></i></a>, <a class="txt" href="ramsey_1.html#E33:14_2_1_12"><i><font color="Green" title="E59">A148</font></i></a>, <a class="txt" href="ramsey_1.html#E2:14_2_1_12"><i><font color="Green" title="E42">A126</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="E42:14_2_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">y</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="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c20">y</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="c19">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="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c20">y</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="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y</font> )</span><br/>

<b>assume </b><a NAME="E1:14_2_1_13"/>
<font color="Maroon" title="c19">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="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c20">y</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="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y</font> )</span><br/>

<a NAME="E2:14_2_1_13"/><b>then </b><i><font color="Green" title="E42">A153</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k'</font>
 
<b>by </b><i><a class="txt" href="ramsey_1.html#E41:14_2_1"><i><font color="Green" title="E41">A125</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:14_2_1_13"/>
<font color="Maroon" title="c20">y</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="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y</font></span><br/>

<b>hence </b><a NAME="E4:14_2_1_13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">y</font>
 <b>by </b><i><a class="txt" href="ramsey_1.html#E2:14_2_1_13"><i><font color="Green" title="E42">A153</font></i></a>, <a class="txt" href="ramsey_1.html#E41:14_2_1"><i><font color="Green" title="E41">A125</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>hence </b><a NAME="E43:14_2_1"/>
<font color="Maroon" title="c7">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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c18">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</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="E3:14_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 ; <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="E3:14"/>
for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="ramsey_1.html#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="ramsey_1.html#E2:14"><i><font color="Green" title="E13">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:14"/>
for <font color="Olive" title="b1">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> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">k</font>  st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <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>  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="b1">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="Olive" title="b2">H</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
