
theorem Th10:
  for x,y,z being Element of REAL holds real_dist.(x,y) <=
  real_dist.(x,z) + real_dist.(z,y)
proof
  let x,y,z be Element of REAL;
  |.x-y.|=|.(x-z)+(z-y).|;
  then
A1: |.x-y.|<=|.x-z.|+|.z-y.| by COMPLEX1:56;
  real_dist.(x,y)=|.x-y.| & real_dist.(x,z)=|.x-z.| by Def12;
  hence thesis by A1,Def12;
end;
