
theorem Th5:
  for M being symmetric triangle Reflexive MetrStruct,
      a, b being Element of M holds 0 <= dist(a,b)
proof
  let M be symmetric triangle Reflexive MetrStruct, a, b be Element of M;
A1: (1/2)*dist(a,a)=(1/2)*0 by Th1;
  dist(a,a)<=dist(a,b)+dist(b,a) & dist(a,b) =(1/2)*(1*dist(a,b)+1*dist(a,
  b)) by Th4;
  hence thesis by A1,XREAL_1:64;
end;
