let h be Real; :: thesis: for f being Function of REAL ,REAL
for n being Element of NAT holds (bdif f,h) . n is Function of REAL ,REAL
let f be Function of REAL ,REAL ; :: thesis: for n being Element of NAT holds (bdif f,h) . n is Function of REAL ,REAL
defpred S₁[ Element of NAT ] means (bdif f,h) . $1 is Function of REAL ,REAL ;
A1: S₁[ 0 ] by Def7;
A2: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume (bdif f,h) . k is Function of REAL ,REAL ; :: thesis: S₁[k + 1]
then bD ((bdif f,h) . k),h is Function of REAL ,REAL by Lm2;
hence S₁[k + 1] by Def7; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A1, A2);
hence for n being Element of NAT holds (bdif f,h) . n is Function of REAL ,REAL ; :: thesis: verum