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 implies (a+b),(a|^2 + b|^2 + a*b) are_coprime
  proof
    assume
    A1: a,b are_coprime;
    A2: |. a|^2 + b|^2 - (1-2)*a*b .| in NAT by INT_1:3;
    (a+b)|^2 gcd (a|^2 + b|^2 + a*b) =
      |. a|^2 + b|^2 - (1-2)*a*b .| gcd |.1.| by A1,Th74
    .= 1 by NEWTON:51,A2; then
    (a+b)|^2, (a|^2 + b|^2 + a*b) are_coprime;
    hence thesis by Th73;
  end;
