reconsider a9 = a, b9 = b as Element of REAL by XREAL_0:def 1;

deffunc H_{1}( Real) -> Element of REAL = In (((a9 * $1) + b9),REAL);

consider f being Function of REAL,REAL such that

A1: for x being Element of REAL holds f . x = H_{1}(x)
from FUNCT_2:sch 4();

take f ; :: thesis: for x being Real holds f . x = (a * x) + b

let x be Real; :: thesis: f . x = (a * x) + b

reconsider x9 = x as Element of REAL by XREAL_0:def 1;

f . x9 = H_{1}(x)
by A1;

hence f . x = (a * x) + b ; :: thesis: verum

deffunc H

consider f being Function of REAL,REAL such that

A1: for x being Element of REAL holds f . x = H

take f ; :: thesis: for x being Real holds f . x = (a * x) + b

let x be Real; :: thesis: f . x = (a * x) + b

reconsider x9 = x as Element of REAL by XREAL_0:def 1;

f . x9 = H

hence f . x = (a * x) + b ; :: thesis: verum