reserve T, T1 for non empty TopSpace;
reserve F,G,H for Subset-Family of T,
  A,B,C,D for Subset of T,
  O,U for open Subset of T,
  p,q for Point of T,
  x,y,X for set;
reserve Un for FamilySequence of T,
  r,r1,r2 for Real,
  n for Element of NAT;
reserve m for Function of [:the carrier of T,the carrier of T:],REAL;

theorem Th29:
  for f be Function of [:X,X:],REAL st f is_a_pseudometric_of X
  for x,y be Element of X holds f.(x,y)>=0
proof
  let f be Function of [:X,X:],REAL such that
A1: f is_a_pseudometric_of X;
  let x,y be Element of X;
  f.(x,x)<=f.(x,y)+f.(y,x) & f.(x,x)=0 by A1,Lm8;
  then 0<=(f.(x,y)+f.(x,y))/2 by A1,Lm8;
  hence thesis;
end;
