let f, f9 be Function of REAL,REAL; :: thesis: ( ( for x being real number holds f . x = (a * x) + b ) & ( for x being real number holds f9 . x = (a * x) + b ) implies f = f9 )
assume that
A2: for x being real number holds f . x = (a * x) + b and
A3: for x being real number holds f9 . x = (a * x) + b ; :: thesis: f = f9

now

let x be Element of REAL ; :: thesis: f . x = f9 . x
thus f . x = (a * x) + b by A2
.= f9 . x by A3 ; :: thesis: verum

end;

hence f = f9 by FUNCT_2:113; :: thesis: verum