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

theorem Th4:
  x <=' y & y <=' x implies x = y
proof
  assume x <=' y;
  then consider z1 being Element of REAL+ such that
A1: x + z1 = y by ARYTM_2:9;
  assume y <=' x;
  then consider z2 being Element of REAL+ such that
A2: y + z2 = x by ARYTM_2:9;
  x = x + (z1 + z2) by A1,A2,ARYTM_2:6;
  then z1 = {} by Th1,ARYTM_2:5;
  hence thesis by A1,ARYTM_2:def 8;
end;
