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 Th76:
  iter(R,m)*iter(R,n) = iter(R,n+m)
proof
  defpred P[Nat] means iter(R,$1)*iter(R,n) = iter(R,n+$1);
A1: P[k] implies P[k+1]
  proof
    assume
A2: iter(R,k)*iter(R,n) = iter(R,n+k);
    thus iter(R,k+1)*iter(R,n) = R*iter(R,k)*iter(R,n) by Th68
      .= R*(iter(R,k)*iter(R,n)) by RELAT_1:36
      .= iter(R,n+k+1) by A2,Th68
      .= iter(R,n+(k+1));
  end;
  iter(R,0)*iter(R,n) = id(field R)*iter(R,n) by Th67
    .= iter(R,n+0) by Th74;
  then
A3: P[ 0];
  P[k] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
