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 Th74:
  a,b are_coprime implies
    (a+b)|^2 gcd (a|^2 + b|^2 - (n-2)*a*b) = a|^2 + b|^2 - (n-2)*a*b gcd n
  proof
    A0: a|^2 = a*a & b|^2 = b*b by NEWTON:81;
    assume
    A1: a,b are_coprime;
    A2: (a+b)|^2 = (a+b)*(a+b) by NEWTON:81
    .= (a|^2+b|^2 - (n-2)*a*b) + n*a*b by A0;
    a|^(0+1)+b|^(0+1),a*b are_coprime by A1,Th27; then
    (a+b)|^2,a*b are_coprime by WSIERP_1:10; then
    1 = (a+b)|^2 - n*(a*b) gcd a*b by Th5
    .=(a|^2+b|^2 -(n-2)*a*b) + n*a*b - n*a*b gcd a*b by A2; then
    A4: (a|^2+b|^2 -(n-2)*a*b),(a*b) are_coprime;
    (a+b)|^2 gcd (a|^2+b|^2 -(n-2)*a*b) =
    n*a*b+1*(a|^2+b|^2 -(n-2)*a*b) gcd (a|^2+b|^2 -(n-2)*a*b) by A2
    .= n*(a*b) gcd (a|^2+b|^2 -(n-2)*a*b) by Th5
    .= n gcd (a|^2+b|^2 - (n-2)*a*b) by A4,INT_6:19;
    hence thesis;
  end;
