let F1, F2 be Subset-Family of (S *); :: thesis: ( ( for A being Subset of (S *) holds
( A in F1 iff ex s being FinSequence of S st A = Finseq-EQclass s ) ) & ( for A being Subset of (S *) holds
( A in F2 iff ex s being FinSequence of S st A = Finseq-EQclass s ) ) implies F1 = F2 )
assume that
A1: for A being Subset of (S *) holds
( A in F1 iff S₁[A] ) and
A2: for A being Subset of (S *) holds
( A in F2 iff S₁[A] ) ; :: thesis: F1 = F2
thus F1 = F2 from SUBSET_1:sch 4(A1, A2); :: thesis: verum