reserve x,y,z,w for ExtReal,
  r for Real;
reserve f,g for ExtReal;

theorem
  z is real implies x = x + z - z
proof
  assume
A1: z is real;
  per cases by XXREAL_0:14;
  suppose
A2: x = +infty;
    then x + z = +infty by A1,Def2;
    hence thesis by A1,A2,Th13;
  end;
  suppose
A3: x = -infty;
    then x + z = -infty by A1,Def2;
    hence thesis by A1,A3,Th14;
  end;
  suppose
    x is Element of REAL;
    then reconsider a = x, c = z as Real by A1;
    x + z - z = a + c - c;
    hence thesis;
  end;
end;
