set f = AffineMap a,b;
for x0 being real number st x0 in REAL holds
(AffineMap a,b) . x0 = (a * x0) + b by Def3;
then ( REAL = dom (AffineMap a,b) & (AffineMap a,b) | REAL is continuous ) by FCONT_1:48, FUNCT_2:def 1;
hence AffineMap a,b is continuous by RELAT_1:98; :: thesis: verum