let A, B be ManySortedFunction of ((FreeSort X) #) * the Arity of S,(FreeSort X) * the ResultSort of S; :: thesis: ( ( for o being OperSymbol of S holds A . o = DenOp (o,X) ) & ( for o being OperSymbol of S holds B . o = DenOp (o,X) ) implies A = B )
assume that
A3: for o being OperSymbol of S holds A . o = DenOp (o,X) and
A4: for o being OperSymbol of S holds B . o = DenOp (o,X) ; :: thesis: A = B
for i being object st i in the carrier' of S holds
A . i = B . i

proof

let i be object ; :: thesis: ( i in the carrier' of S implies A . i = B . i )
assume i in the carrier' of S ; :: thesis: A . i = B . i
then reconsider s = i as OperSymbol of S ;
A . s = DenOp (s,X) by A3;
hence A . i = B . i by A4; :: thesis: verum

end;

hence A = B ; :: thesis: verum