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

proof

assume A1: f is complex-valued ; :: thesis: for x being object st x in dom f holds
f . x is complex
let x be object ; :: thesis: ( x in dom f implies f . x is complex )
assume A2: x in dom f ; :: thesis: f . x is complex
reconsider f = f as complex-valued Function by A1;
f . x in rng f by A2, FUNCT_1:3;
hence f . x is complex ; :: thesis: verum

end;

assume A3: for x being object st x in dom f holds
f . x is complex ; :: thesis: f is complex-valued
let y be object ; :: according to TARSKI:def 3,VALUED_0:def 1 :: thesis: ( not y in rng f or y in COMPLEX )
assume y in rng f ; :: thesis: y in COMPLEX
then ex x being object st
( x in dom f & y = f . x ) by FUNCT_1:def 3;
then y is complex by A3;
hence y in COMPLEX by XCMPLX_0:def 2; :: thesis: verum