thus ( f is real-valued implies for x being object st x in dom f holds
f . x is real ) :: thesis: ( ( for x being object st x in dom f holds
f . x is real ) implies f is real-valued )

proof

assume A7: f is real-valued ; :: thesis: for x being object st x in dom f holds
f . x is real
let x be object ; :: thesis: ( x in dom f implies f . x is real )
assume A8: x in dom f ; :: thesis: f . x is real
reconsider f = f as real-valued Function by A7;
f . x in rng f by A8, FUNCT_1:3;
hence f . x is real ; :: thesis: verum

end;

assume A9: for x being object st x in dom f holds
f . x is real ; :: thesis: f is real-valued
let y be object ; :: according to TARSKI:def 3,VALUED_0:def 3 :: thesis: ( not y in rng f or y in REAL )
assume y in rng f ; :: thesis: y in REAL
then ex x being object st
( x in dom f & y = f . x ) by FUNCT_1:def 3;
then y is real by A9;
hence y in REAL by XREAL_0:def 1; :: thesis: verum