let f, g be ManySortedFunction of ((OSClass R) #) * the Arity of S,(OSClass R) * the ResultSort of S; :: thesis: ( ( for o being OperSymbol of S holds f . o = OSQuotCharact (R,o) ) & ( for o being OperSymbol of S holds g . o = OSQuotCharact (R,o) ) implies f = g )
assume that
A3: for o being OperSymbol of S holds f . o = OSQuotCharact (R,o) and
A4: for o being OperSymbol of S holds g . o = OSQuotCharact (R,o) ; :: thesis: f = g

now :: thesis: for i being object st i in the carrier' of S holds
f . i = g . i

let i be object ; :: 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 = OSQuotCharact (R,i1) by A3;
hence f . i = g . i by A4; :: thesis: verum

end;

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