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 Th28:
  a,b are_coprime implies a*b,(a|^(n+1) - b|^(n+1)) are_coprime
  proof
    A1: b*b|^n = |.b|^(n+1).| by NEWTON:6;
    A2: a*a|^n = |.a|^(n+1).| by NEWTON:6;
    A3: |.(a|^(n+1)-b|^(n+1)).| =  |.-(a|^(n+1)-b|^(n+1)).| by COMPLEX1:52;
    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 A1,A2,A3,INT_2:34; then
    |.a|^(n+1)-b|^(n+1).|, b are_coprime & |.a|^(n+1)-b|^(n+1).|, a
      are_coprime by NEWTON01:41; then
    |.a|^(n+1)-b|^(n+1).|,a*b are_coprime by INT_2:26;
    hence thesis by INT_6:14;
  end;
