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) ;
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 MEASURE6:60;
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, PARTFUN1:34; :: 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