let W1, W2 be OSSubset of OU0; :: thesis: ( ( for s being SortSymbol of S1 holds W1 . s = meet (OSSubSort (A,s)) ) & ( for s being SortSymbol of S1 holds W2 . s = meet (OSSubSort (A,s)) ) implies W1 = W2 )
assume that
A5: for s being SortSymbol of S1 holds W1 . s = meet (OSSubSort (A,s)) and
A6: for s being SortSymbol of S1 holds W2 . s = meet (OSSubSort (A,s)) ; :: thesis: W1 = W2
for s being object st s in the carrier of S1 holds
W1 . s = W2 . s

proof

let s be object ; :: 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 s = s as SortSymbol of S1 ;
W1 . s = meet (OSSubSort (A,s)) by A5;
hence W1 . s = W2 . s by A6; :: thesis: verum

end;

hence W1 = W2 ; :: thesis: verum