deffunc H₁( Real) -> set = (h ") . (f . (h . $1));
A1: for a being Element of [.0,1.] holds H₁(a) in [.0,1.]

proof

let a be Element of [.0,1.]; :: thesis: H₁(a) in [.0,1.]
(h ") . (f . (h . a)) in [.0,1.] ;
hence H₁(a) in [.0,1.] ; :: thesis: verum

end;

ex f being UnOp of [.0,1.] st
for a being Element of [.0,1.] holds f . a = H₁(a) from FUNCT_2:sch 8(A1);
hence ex b₁ being UnOp of [.0,1.] st
for x being Element of [.0,1.] holds b₁ . x = (h ") . (f . (h . x)) ; :: thesis: verum