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 ;
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 ;
take H ; :: thesis: ( H is infinite & H is_homogeneous_for P )
thus H is infinite by A5; :: thesis:
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 ;
then h in (dom F) /\ (the_subsets_of_card (n,H)) by ;
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 ;
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
.= (F | (the_subsets_of_card (n,H))) . h by
.= F . h by ;
then F1 . ((proj P) . x9) = (F1 * (proj P)) . h by
.= F1 . ((proj P) . h) by ;
then ( h in E & (proj P) . h = (proj P) . x9 ) by ;
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