let W1, W2 be OSSubset of OU0; :: thesis: ( ( for s being SortSymbol of S1 holds W1 . s = OSConstants OU0,s ) & ( for s being SortSymbol of S1 holds W2 . s = OSConstants OU0,s ) implies W1 = W2 )
assume that
A6: for s being SortSymbol of S1 holds W1 . s = OSConstants OU0,s and
A7: for s being SortSymbol of S1 holds W2 . s = OSConstants OU0,s ; :: thesis: W1 = W2
for s being set st s in the carrier of S1 holds
W1 . s = W2 . s

proof

let s be set ; :: thesis: ( s in the carrier of S1 implies W1 . s = W2 . s )
assume s in the carrier of S1 ; :: thesis: W1 . s = W2 . s
then reconsider t = s as SortSymbol of S1 ;
W1 . t = OSConstants OU0,t by A6;
hence W1 . s = W2 . s by A7; :: thesis: verum

end;

hence W1 = W2 by PBOOLE:3; :: thesis: verum