thus ( f is rational-functions-valued implies for x being object st x in dom f holds
f . x is RAT -valued Function ) :: thesis: ( ( for x being object st x in dom f holds
f . x is RAT -valued Function ) implies f is rational-functions-valued )
proof
assume A7: rng f is rational-functions-membered ; :: according to VALUED_2:def 23 :: thesis: for x being object st x in dom f holds
f . x is RAT -valued Function

let x be object ; :: thesis: ( x in dom f implies f . x is RAT -valued Function )
assume x in dom f ; :: thesis:
then f . x in rng f by FUNCT_1:def 3;
hence f . x is RAT -valued Function by A7; :: thesis: verum
end;
assume A8: for x being object st x in dom f holds
f . x is RAT -valued Function ; :: thesis:
let y be object ; :: according to VALUED_2:def 5,VALUED_2:def 23 :: thesis: ( y in rng f implies y is RAT -valued Function )
assume y in rng f ; :: thesis:
then ex x being object st
( x in dom f & f . x = y ) by FUNCT_1:def 3;
hence y is RAT -valued Function by A8; :: thesis: verum