hereby :: thesis: ( ( for x being set st x in dom f holds
f . x is TolStr ) implies f is TolStr-yielding )

assume A1: f is TolStr-yielding ; :: thesis: for x being set st x in dom f holds
f . x is TolStr
let x be set ; :: thesis: ( x in dom f implies f . x is TolStr )
assume x in dom f ; :: thesis: f . x is TolStr
then f . x in rng f by FUNCT_1:12;
hence f . x is TolStr by A1, Def13; :: thesis: verum

end;

assume A2: for x being set st x in dom f holds
f . x is TolStr ; :: thesis: f is TolStr-yielding
let P be set ; :: according to PCS_0:def 13 :: thesis: ( P in rng f implies P is TolStr )
assume P in rng f ; :: thesis: P is TolStr
then ex x being set st
( x in dom f & f . x = P ) by FUNCT_1:def 5;
hence P is TolStr by A2; :: thesis: verum