let f1, f2 be EqualSet of S; :: thesis: ( ( for s being SortSymbol of S holds f1 . s = { e where e is Element of (Equations S) . s : A |= e } ) & ( for s being SortSymbol of S holds f2 . s = { e where e is Element of (Equations S) . s : A |= e } ) implies f1 = f2 )
assume that
A3: for s being SortSymbol of S holds f1 . s = { e where e is Element of (Equations S) . s : A |= e } and
A4: for s being SortSymbol of S holds f2 . s = { e where e is Element of (Equations S) . s : A |= e } ; :: thesis: f1 = f2

now :: thesis: for x being object st x in the carrier of S holds
f1 . x = f2 . x

let x be object ; :: thesis: ( x in the carrier of S implies f1 . x = f2 . x )
assume x in the carrier of S ; :: thesis: f1 . x = f2 . x
then reconsider s = x as SortSymbol of S ;
thus f1 . x = { e where e is Element of (Equations S) . s : A |= e } by A3
.= f2 . x by A4 ; :: thesis: verum

end;

hence f1 = f2 ; :: thesis: verum