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
  for a,b st a,b are_coprime holds for n be non zero Nat holds
  a*b = c|^n iff
    (n-root a in NAT & n-root b in NAT & c = (n-root a)*(n-root b))
  proof
    let a,b such that
A1: a,b are_coprime;
    let n be non zero Nat;
    thus a*b = c|^n implies
      (n-root a in NAT & n-root b in NAT & c = (n-root a)*(n-root b))
    proof
      assume
  B1: a*b = c|^n; then
      consider k such that
  B2: a = k|^n by A1,NEWTON02:30;
      consider l such that
  B3: b = l|^n by A1,B1, NEWTON02:30;
      n-root c|^n = n-root (a*b) by B1
      .= (n-root a)*(n-root b) by POWER:11,NAT_1:14
      .=(n-root k|^n)*(n-root l|^n) by B2,B3;
      hence thesis by ORDINAL1:def 12,B2,B3;
    end;
    assume
    n-root a in NAT & n-root b in NAT & c = (n-root a)*(n-root b); then
    c|^n = (n-root a)|^n*(n-root b)|^n by NEWTON:7;
    hence thesis;
  end;
