reconsider a9 = a, b9 = b as Element of REAL by XREAL_0:def 1;
deffunc H₁( 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 = H₁(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 = H₁(x) by A1;
hence f . x = (a * x) + b ; :: thesis: verum