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

assume A3: f is pcs-yielding ; :: thesis: for x being set st x in dom f holds
f . x is pcs
let x be set ; :: thesis: ( x in dom f implies f . x is pcs )
assume x in dom f ; :: thesis: f . x is pcs
then f . x in rng f by FUNCT_1:3;
hence f . x is pcs by A3, Def29; :: thesis: verum

end;

assume A4: for x being set st x in dom f holds
f . x is pcs ; :: thesis: f is pcs-yielding
let P be set ; :: according to PCS_0:def 29 :: thesis: ( P in rng f implies P is pcs )
assume P in rng f ; :: thesis: P is pcs
then ex x being set st
( x in dom f & f . x = P ) by FUNCT_1:def 3;
hence P is pcs by A4; :: thesis: verum