reserve x,y for Element of REAL;
reserve i,j,k for Element of NAT;
reserve a,b for Element of REAL;

theorem Th11:
  for x,o being Element of REAL st o = 0 holds +(x,o) = x
proof
  reconsider y9 = 0 as Element of REAL+ by ARYTM_2:20;
  let x,o being Element of REAL such that
A1: o = 0;
  per cases;
  suppose
    x in REAL+;
    then reconsider x9 = x as Element of REAL+;
    x9 = x9 + y9 by ARYTM_2:def 8;
    hence thesis by A1,Def1;
  end;
  suppose
A2: not x in REAL+;
    x in REAL+ \/ [:{{}},REAL+:] by XBOOLE_0:def 5;
    then
A3: x in [:{{}},REAL+:] by A2,XBOOLE_0:def 3;
    then consider x1,x2 being object such that
A4: x1 in {{}} and
A5: x2 in REAL+ and
A6: x = [x1,x2] by ZFMISC_1:84;
    reconsider x9 = x2 as Element of REAL+ by A5;
A7: x1 = 0 by A4,TARSKI:def 1;
    then x = y9 - x9 by A6,Th3,ARYTM_1:19;
    hence thesis by A1,A3,A6,A7,Def1;
  end;
end;
