reserve r,s,t,x9,y9,z9,p,q for Element of RAT+;
reserve x,y,z for Element of REAL+;

theorem Th5:
  x + y = {} implies x = {}
proof
  assume
A1: x + y = {};
  per cases;
  suppose
    y = {};
    hence thesis by A1,Def8;
  end;
  suppose
A2: y <> {};
    then DEDEKIND_CUT y <> {} by Lm10;
    then consider s0 being Element of RAT+ such that
A3: s0 in DEDEKIND_CUT y by SUBSET_1:4;
    assume
A4: x <> {};
    then DEDEKIND_CUT x <> {} by Lm10;
    then consider r0 being Element of RAT+ such that
A5: r0 in DEDEKIND_CUT x by SUBSET_1:4;
A6: r0 + s0 in { r + s : r in DEDEKIND_CUT x & s in DEDEKIND_CUT y } by A5,A3;
    GLUED(DEDEKIND_CUT x + DEDEKIND_CUT y) = {} by A1,A2,A4,Def8;
    hence contradiction by A6,Lm11;
  end;
end;
