reserve k,n,m,l,p for Nat;
reserve n0,m0 for non zero Nat;
reserve f for FinSequence;
reserve x,X,Y for set;
reserve f1,f2,f3 for FinSequence of REAL;
reserve n1,n2,m1,m2 for Nat;
reserve I,j for set;
reserve f,g for Function of I, NAT;
reserve J,K for finite Subset of I;

theorem Th35:
  EXP(k) is multiplicative
proof
  for n,m being non zero Nat st n,m are_coprime holds (
  EXP(k)).(n*m) = (EXP(k)).n * (EXP(k)).m
  proof
    let n,m be non zero Nat;
    assume n,m are_coprime;
    thus (EXP(k)).(n*m) = (n*m)|^k by Def1
      .= n|^k * m|^k by NEWTON:7
      .= (EXP(k)).n * m|^k by Def1
      .= (EXP(k)).n * (EXP(k)).m by Def1;
  end;
  hence EXP(k) is multiplicative;
end;
