let f be Function; :: thesis: for n being Nat holds iter f,(n + 1) = f * (iter f,n)
let n be Nat; :: thesis: iter f,(n + 1) = f * (iter f,n)
defpred S₁[ Nat] means iter f,($1 + 1) = f * (iter f,$1);
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: iter f,(k + 1) = f * (iter f,k) ; :: thesis: S₁[k + 1]
thus f * (iter f,(k + 1)) = f * ((iter f,k) * f) by Th71
.= (f * (iter f,k)) * f by RELAT_1:55
.= iter f,((k + 1) + 1) by A2, Th71 ; :: thesis: verum

end;

iter f,(0 + 1) = f by Th72
.= f * (id ((dom f) \/ (rng f))) by Lm3
.= f * (iter f,0 ) by Th70 ;
then A3: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A1);
hence iter f,(n + 1) = f * (iter f,n) ; :: thesis: verum