reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem
  for a,b be odd Nat st a <> b holds
  1 = min (2 |-count (a-b), 2|-count (a+b))
  proof
    let a,b be odd Nat such that
    A0: a <> b;
    reconsider k = |.a-b.| as even Nat;
    reconsider l = a+b as even Nat;
    A1: 2|^2 = 2*2 by NEWTON:81;
    A2: 2|^1 divides |.a-b.| & 2|^1 divides (a+b) by ABIAN:def 1;
    A3: a - b <> b - b by A0; then
    A4: 2 |-count |.a-b.| <> 0 & 2 |-count (a+b) <> 0 by A2,NAT_3:27;
    per cases;
    suppose
      not 4 divides (a - b); then
      not |.2|^(1+1).| divides |.a-b.| by A1,INT_2:16; then
      1 = 2 |-count |.a-b.| by A2,A3,NAT_3:def 7;
      hence thesis by A4,NAT_1:14,XXREAL_0:def 9;
    end;
    suppose 4 divides (a-b); then
      not 2|^(1+1) divides (a+b) by A1,NEWTON02:58; then
      2 |-count (a+b) = 1 by A2,NAT_3:def 7;
      hence thesis by A4,NAT_1:14,XXREAL_0:def 9;
    end;
  end;
