reserve a,b,c for Integer;
reserve i,j,k,l for Nat;
reserve n for Nat;
reserve a,b,c,d,a1,b1,a2,b2,k,l for Integer;
reserve p,p1,q,l for Nat;

theorem
  a gcd b = |.a.| gcd |.b.|
proof
A1: |.b.| = b or |.b.| = -b by ABSVALUE:1;
A2: |.a.| = a or |.a.| = -a by ABSVALUE:1;
A3: now
    let m be Integer;
    assume m divides |.a.| & m divides |.b.|;
    then m divides a & m divides b by A2,A1,Th10;
    hence m divides a gcd b by Def2;
  end;
  a gcd b divides b by Def2;
  then
A4: a gcd b divides |.b.| by A1,Th10;
  a gcd b divides a by Def2;
  then a gcd b divides |.a.| by A2,Th10;
  hence thesis by A4,A3,Def2;
end;
