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
  for X being strict non empty MetrSpace holds the distance of X is
  Reflexive discerning symmetric triangle
proof
  let X be strict non empty MetrSpace;
A1: the distance of X is_metric_of the carrier of X by PCOMPS_1:35;
  hence the distance of X is Reflexive by Th3;
  thus the distance of X is discerning by A1,Th3;
  thus the distance of X is symmetric by A1,Th3;
  thus thesis by A1,Th3;
end;
