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 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 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 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 Nat holds g . (h,(n + 1)) = (g . (h,n)) * h ) ) ; :: thesis: f = g

now :: thesis: for h being Element of X
for n being Nat holds f . (h,n) = g . (h,n)

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

A12: now :: thesis: for n being Nat st S₁[n] holds
S₁[n + 1]

let n be 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 Nat holds S₁[n] from NAT_1:sch 2(A15, A12);
hence f . (h,n) = g . (h,n) ; :: thesis: verum

end;

hence f = g ; :: thesis: verum