reserve x,y,z for Element of REAL+;

theorem Th3:
  x <=' y & y <=' z implies x <=' z
proof
  assume x <=' y;
  then consider z1 being Element of REAL+ such that
A1: x + z1 = y by ARYTM_2:9;
  assume y <=' z;
  then consider z2 being Element of REAL+ such that
A2: y + z2 = z by ARYTM_2:9;
  z = x + (z1 + z2) by A1,A2,ARYTM_2:6;
  hence thesis by ARYTM_2:19;
end;
