let n, k be Nat; :: thesis: for X being infinite set
for P being a_partition of the_subsets_of_card (n,X) st card P = k holds
ex H being Subset of X st
( H is infinite & H is_homogeneous_for P )
let X be infinite set ; :: thesis: for P being a_partition of the_subsets_of_card (n,X) st card P = k holds
ex H being Subset of X st
( H is infinite & H is_homogeneous_for P )
let P be a_partition of the_subsets_of_card (n,X); :: thesis: ( card P = k implies ex H being Subset of X st
( H is infinite & H is_homogeneous_for P ) )
assume A1: card P = k ; :: thesis: ex H being Subset of X st
( H is infinite & H is_homogeneous_for P )
then P,k are_equipotent by CARD_1:def 2;
then consider F1 being Function such that
A2: F1 is one-to-one and
A3: dom F1 = P and
A4: rng F1 = k by WELLORD2:def 4;
reconsider F1 = F1 as Function of P,k by A3, A4, FUNCT_2:1;
set F = F1 * (proj P);
reconsider F = F1 * (proj P) as Function of (the_subsets_of_card (n,X)),k ;
consider H being Subset of X such that
A5: H is infinite and
A6: F | (the_subsets_of_card (n,H)) is constant by A1, Th14;
take H ; :: thesis: ( H is infinite & H is_homogeneous_for P )
thus H is infinite by A5; :: thesis: H is_homogeneous_for P
set h = the Element of the_subsets_of_card (n,H);
a7: not the_subsets_of_card (n,H) is empty by A5;
A8: the_subsets_of_card (n,H) c= the_subsets_of_card (n,X) by Lm1;
then reconsider h = the Element of the_subsets_of_card (n,H) as Element of the_subsets_of_card (n,X) by a7;
set E = EqClass (h,P);
reconsider E = EqClass (h,P) as Element of P by EQREL_1:def 6;
for x being object st x in the_subsets_of_card (n,H) holds
x in E

proof

let x be object ; :: thesis: ( x in the_subsets_of_card (n,H) implies x in E )
assume A9: x in the_subsets_of_card (n,H) ; :: thesis: x in E
then reconsider x9 = x as Element of the_subsets_of_card (n,X) by A8;
A10: dom F = the_subsets_of_card (n,X) by A1, FUNCT_2:def 1;
then h in (dom F) /\ (the_subsets_of_card (n,H)) by A5, XBOOLE_0:def 4;
then A11: h in dom (F | (the_subsets_of_card (n,H))) by RELAT_1:61;
x9 in (dom F) /\ (the_subsets_of_card (n,H)) by A9, A10, XBOOLE_0:def 4;
then A12: x9 in dom (F | (the_subsets_of_card (n,H))) by RELAT_1:61;
F . x9 = (F | (the_subsets_of_card (n,H))) . x9 by A9, FUNCT_1:49
.= (F | (the_subsets_of_card (n,H))) . h by A6, A12, A11, FUNCT_1:def 10
.= F . h by A5, FUNCT_1:49 ;
then F1 . ((proj P) . x9) = (F1 * (proj P)) . h by A10, FUNCT_1:12
.= F1 . ((proj P) . h) by A10, FUNCT_1:12 ;
then ( h in E & (proj P) . h = (proj P) . x9 ) by A2, A3, EQREL_1:def 6, FUNCT_1:def 4;
hence x in E by Th13; :: thesis: verum

end;

then the_subsets_of_card (n,H) c= E ;
hence H is_homogeneous_for P ; :: thesis: verum