reserve a,x,y for object, A,B for set,
  l,m,n for Nat;
reserve X,Y for set, x for object,
  p,q for Function-yielding FinSequence,
  f,g,h for Function;
reserve m,n,k for Nat, R for Relation;

theorem
  n <> 0 implies iter(iter(R,m),n) = iter(R,m*n)
proof
  defpred P[Nat] means iter(iter(R,m),$1+1) = iter(R,m*($1+1));
A1: P[k] implies P[k+1] by Lm5;
A2: P[ 0] by Th69;
A3: P[k] from NAT_1:sch 2(A2,A1);
  assume n <> 0;
  then consider k be Nat such that
A4: n = k+1 by NAT_1:6;
  reconsider k as Nat;
  n=k+1 by A4;
  hence thesis by A3;
end;
