reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem Th7:
  a|^n gcd b|^n = (a gcd b)|^n
  proof
    a gcd b = k implies a|^n gcd b|^n = k|^n
    proof
      assume
      A0: a gcd b = k; then
      consider l be Nat such that
      A2: a = k * l by INT_2:21,NAT_D:def 3;
      consider m be Nat such that
      A3: b = k * m by A0,INT_2:21,NAT_D:def 3;
      per cases;
      suppose
        A4: a > 0; then
        l gcd m = 1 by A0,A2,A3,Lm6; then
        A6: l|^n gcd m|^n = 1 by WSIERP_1:12;
a10:    l > 0 by A2,A4;
        k|^n = (k|^n)*(l|^n gcd m|^n) by A6
        .= (k|^n)*(l|^n) gcd (k|^n)*(m|^n) by a10,EULER_1:15
        .= k|^n*l|^n gcd (k*m)|^n by NEWTON:7
        .= a|^n gcd b|^n by A2,A3,NEWTON:7;
        hence thesis;
      end;
      suppose
        A14: a = 0; then
        A16: a|^n = 0 or n < 1 by NEWTON:11;
        n = 0 implies a|^n = 1 & b|^n=1 & k|^n = 1 by NEWTON:4;
        hence thesis by A0,A14,A16,NAT_1:14,NAT_D:32;
      end;
    end;
    hence thesis;
  end;
