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 is even & a,b are_coprime implies (a-b),(a+b) are_coprime
  proof
    assume
    A1: a is even & a,b are_coprime; then
    not 2 divides (a gcd b) by NAT_D:7; then
    A2: b is odd by A1,INT_2:def 2; then
    A3: ((a+b) gcd (a-b)) is odd by A1,Th9;
    A4: (a+b) gcd (a-b) <> 0 by A2,INT_2:5;
    per cases by A1,A2,XXREAL_0:1;
    suppose
      a > b; then
      (a+b) gcd (a-b) <= 2 by A1,Th8; then
      (a+b) gcd (a-b) < 2 by A3,XXREAL_0:1;
      hence thesis by A4,NAT_1:23;
    end;
    suppose
      a < b; then
      (a+b) gcd (b-a) <= 2 by A1,Th8; then
      (a+b) gcd -(b-a) <= 2 by Th1; then
      (a+b) gcd (a-b) < 2 by A3,XXREAL_0:1;
      hence thesis by A4,NAT_1:23;
    end;
  end;
