let f, g be ManySortedFunction of (the Sorts of A # ) * the Arity of S,((OSClass R) # ) * the Arity of S; :: thesis: ( ( for o being OperSymbol of S holds f . o = OSQuotArgs R,o ) & ( for o being OperSymbol of S holds g . o = OSQuotArgs R,o ) implies f = g )
assume that
A7: for o being OperSymbol of S holds f . o = OSQuotArgs R,o and
A8: for o being OperSymbol of S holds g . o = OSQuotArgs R,o ; :: thesis: f = g

now

let i be set ; :: thesis: ( i in the carrier' of S implies f . i = g . i )
assume i in the carrier' of S ; :: thesis: f . i = g . i
then reconsider i1 = i as OperSymbol of S ;
f . i1 = OSQuotArgs R,i1 by A7;
hence f . i = g . i by A8; :: thesis: verum

end;

hence f = g by PBOOLE:3; :: thesis: verum