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
  a,b are_coprime & a*b = c|^n implies ex k st k|^n = a
  proof
    assume
    A1: a,b are_coprime & a*b = c|^n;
    consider k such that
    A3: k = a gcd c;
    per cases;
    suppose
      B1: n = 0; then
      a = 1|^0 by A1,NAT_1:15,NEWTON:4;
      hence thesis by B1;
    end;
    suppose
      n> 0 & a = 0; then
      a = a|^n by NEWTON:11,NAT_1:14;
      hence thesis;
    end;
    suppose b = 0; then
      a = 1|^n by A1;
      hence thesis;
    end;
    suppose
      B1: n > 0 & a > 0 & b > 0; then
      consider m such that
      B2: n = 1 + m by NAT_1:10,14;
      B3: a|^m,b are_coprime by A1,WSIERP_1:10;
      k|^n = a|^n gcd c|^n by A3,Th7
      .= a|^m*a gcd a*b by A1,B2,NEWTON:6
      .= a*1 by B3,B1,EULER_1:15;
      hence thesis;
    end;
  end;
