let f, g be Function of [:the carrier of X,NAT :],the carrier of X; :: thesis: ( ( for h being Element of X holds
( f . h,0 = 0. X & ( for n being Element of NAT holds f . h,(n + 1) = (f . h,n) * h ) ) ) & ( for h being Element of X holds
( g . h,0 = 0. X & ( for n being Element of NAT holds g . h,(n + 1) = (g . h,n) * h ) ) ) implies f = g )
assume that
A10: for h being Element of X holds
( f . h,0 = 0. X & ( for n being Element of NAT holds f . h,(n + 1) = (f . h,n) * h ) ) and
A11: for h being Element of X holds
( g . h,0 = 0. X & ( for n being Element of NAT holds g . h,(n + 1) = (g . h,n) * h ) ) ; :: thesis: f = g

now

let h be Element of X; :: thesis: for n being Element of NAT holds f . h,n = g . h,n
let n be Element of NAT ; :: thesis: f . h,n = g . h,n
defpred S₁[ Element of NAT ] means f . [h,$1] = g . [h,$1];

A12: now

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A13: S₁[n] ; :: thesis: S₁[n + 1]
A14: g . [h,n] = g . h,n ;
f . [h,(n + 1)] = f . h,(n + 1)
.= (f . h,n) * h by A10
.= g . h,(n + 1) by A11, A13, A14
.= g . [h,(n + 1)] ;
hence S₁[n + 1] ; :: thesis: verum

end;

f . [h,0 ] = f . h,0
.= 0. X by A10
.= g . h,0 by A11
.= g . [h,0 ] ;
then A15: S₁[ 0 ] ;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A15, A12);
hence f . h,n = g . h,n ; :: thesis: verum

end;

hence f = g by BINOP_1:2; :: thesis: verum