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;
