deffunc H₁( Nat) -> Subset of X = Union (F ^\ $1);
consider f being SetSequence of X such that
A6: for n being Element of NAT holds f . n = H₁(n) from FUNCT_2:sch 4();
take meet f ; :: thesis: ex f being SetSequence of X st
( meet f = meet f & ( for n being Nat holds f . n = Union (F ^\ n) ) )
take f ; :: thesis: ( meet f = meet f & ( for n being Nat holds f . n = Union (F ^\ n) ) )
thus meet f = meet f ; :: thesis: for n being Nat holds f . n = Union (F ^\ n)
let n be Nat; :: thesis: f . n = Union (F ^\ n)
n in NAT by ORDINAL1:def 12;
hence f . n = Union (F ^\ n) by A6; :: thesis: verum