reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem LmLCM:
  for a,b be Integer holds a|^n lcm b|^n = (a lcm b)|^n
proof
  let a,b be Integer;
  A2: a|^n gcd b|^n = (a gcd b)|^n by NEWTON027;
  A3: |.a|^n*b|^n.| = (a|^n gcd b|^n)*(a|^n lcm b|^n) &
    |.a*b.| = (a gcd b)*(a lcm b) by NATD29; then
  A4: (a|^n gcd b|^n)*(a|^n lcm b|^n) = |.(a*b)|^n.| by NEWTON:7
  .= ((a gcd b)*(a lcm b))|^n by A3,TAYLOR_2:1
  .= (a gcd b)|^n*(a lcm b)|^n by NEWTON:7
  .= (a|^n gcd b|^n)*(a lcm b)|^n by NEWTON027;
  per cases;
  suppose (a gcd b) = 0; then
    a = 0 & b = 0;
    hence thesis;
  end;
  suppose (a gcd b) <> 0;
    hence thesis by A2,A4,XCMPLX_1:5;
  end;
end;
