reserve X for MetrSpace,
  x,y,z for Element of X,
  A for non empty set,
  G for Function of [:A,A:],REAL,
  f for Function,
  k,n,m,m1,m2 for Nat,
  q,r for Real;

theorem Th2:
  G is_metric_of A implies for a,b being Element of A holds 0 <= G.(a,b)
proof
  assume
A1: G is_metric_of A;
  let a,b be Element of A;
A2: (1/2)*G.(a,a) = (1/2)*0 by A1,PCOMPS_1:def 6;
  G.(a,b) = (1/2)*(1*G.(a,b) + G.(a,b))
    .= (1/2)*(G.(a,b) + G.(b,a)) by A1,PCOMPS_1:def 6;
  hence thesis by A1,A2,PCOMPS_1:def 6;
end;
