reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;
reserve r, u for Real,
  k for Nat;

theorem
  for a,b be Integer st a>=0 & b>=0 holds a gcd b = a gcd (b-a)
proof
  let a,b be Integer;
  assume that
A1: a>=0 and
A2: b>=0;
  thus a gcd b=|.a.| gcd |.b.| by INT_2:34
    .=|.a.| gcd |.|.b.|-|.a.|.| by Th96
    .=|.a.| gcd |.b-|.a.|.| by A2,ABSVALUE:def 1
    .=|.a.| gcd |.b-a.| by A1,ABSVALUE:def 1
    .=a gcd (b-a) by INT_2:34;
end;
