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

theorem Th9:
  x <=' y implies ex z st x + z = y
proof
  assume
A1: x <=' y;
  per cases;
  suppose
A2: x = {};
    take y;
    thus thesis by A2,Def8;
  end;
  suppose
A3: x = y;
    reconsider z = {} as Element of REAL+ by Th1;
    take z;
    thus thesis by A3,Def8;
  end;
  suppose that
A4: x <> {} and
A5: x <> y;
A6: DEDEKIND_CUT x <> {} by A4,Lm10;
    DEDEKIND_CUT x <> DEDEKIND_CUT y by A5,Lm22;
    then consider C being Element of DEDEKIND_CUTS such that
A7: DEDEKIND_CUT x + C = DEDEKIND_CUT y by A1,A6,Lm36,Lm37;
    take GLUED C;
    now
      assume
A8:   C = {};
      not ex e being object st e in { r + s : r in C & s in DEDEKIND_CUT x}
      proof
        given e being object such that
A9:     e in { r + s : r in C & s in DEDEKIND_CUT x};
        ex r,s st e = r + s & r in C & s in DEDEKIND_CUT x by A9;
        hence contradiction by A8;
      end;
      then { r + s : r in C & s in DEDEKIND_CUT x} = {} by XBOOLE_0:def 1;
      then DEDEKIND_CUT y = {} by A7,Def6;
      hence contradiction by A1,A6,Lm37,XBOOLE_1:3;
    end;
    then GLUED C <> {} by Lm11;
    hence x + GLUED C = GLUED(DEDEKIND_CUT x + DEDEKIND_CUT GLUED C) by A4,Def8
      .= GLUED DEDEKIND_CUT y by A7,Lm12
      .= y by Lm23;
  end;
end;
