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 Th27:
  a,b are_coprime implies a*b,(a|^(n+1) + b|^(n+1)) are_coprime
  proof
    assume a,b are_coprime; then
    a|^(n+1), b|^(n+1) are_coprime by WSIERP_1:11; then
    (a|^(n+1)+1*b|^(n+1)) gcd b|^(n+1) = 1 &
      (b|^(n+1)+1*a|^(n+1)) gcd a|^(n+1) = 1 by Th5; then
    (a|^(n+1)+b|^(n+1))*1, b*b|^n are_coprime &
      (a|^(n+1)+b|^(n+1))*1, a*a|^n are_coprime by NEWTON:6; then
    (a|^(n+1)+b|^(n+1)), b are_coprime & (a|^(n+1)+b|^(n+1)), a are_coprime
      by NEWTON01:41;
    hence thesis by INT_2:26;
  end;
