for IT being Function of REAL,REAL holds
( IT = Shift (f,h) iff for x being Real holds IT . x = f . (x + h) )

proof

let IT be Function of REAL,REAL; :: thesis: ( IT = Shift (f,h) iff for x being Real holds IT . x = f . (x + h) )

hereby :: thesis: ( ( for x being Real holds IT . x = f . (x + h) ) implies IT = Shift (f,h) )

assume A1: IT = Shift (f,h) ; :: thesis: for x being Real holds IT . x = f . (x + h)
let x be Real; :: thesis: IT . x = f . (x + h)
dom (Shift (f,h)) = REAL by A1, FUNCT_2:def 1;
then x in dom (Shift (f,h)) by XREAL_0:def 1;
then x in (- h) ++ (dom f) by Def1;
hence IT . x = f . (x + h) by A1, Def1; :: thesis: verum

end;

( dom (Shift (f,h)) = (- h) ++ (dom f) & dom f = REAL ) by Def1, FUNCT_2:def 1;
then dom (Shift (f,h)) = REAL by MEASURE6:24;
then A2: dom IT = dom (Shift (f,h)) by FUNCT_2:def 1;
assume A3: for x being Real holds IT . x = f . (x + h) ; :: thesis: IT = Shift (f,h)
for x being Element of REAL st x in dom IT holds
IT . x = (Shift (f,h)) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom IT implies IT . x = (Shift (f,h)) . x )
assume x in dom IT ; :: thesis: IT . x = (Shift (f,h)) . x
then A4: x in (- h) ++ (dom f) by A2, Def1;
IT . x = f . (x + h) by A3;
hence IT . x = (Shift (f,h)) . x by A4, Def1; :: thesis: verum

end;

hence IT = Shift (f,h) by A2, PARTFUN1:5; :: thesis: verum

end;

hence for b₁ being Function of REAL,REAL holds
( b₁ = Shift (f,h) iff for x being Real holds b₁ . x = f . (x + h) ) ; :: thesis: verum