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

theorem Th22:
  x is real implies y - x + x = y & y + x - x = y
proof
  assume
A1: x is real;
  per cases by A1,XXREAL_0:14;
  suppose
    x in REAL & y in REAL;
    then reconsider a = x, b = y as Real;
A2: (y + x) - x = b + a - a .= y;
    y - x + x = (b - a) + a .= y;
    hence thesis by A2;
  end;
  suppose
A3: x in REAL & y = -infty;
    then y - x = -infty by Def2;
    hence thesis by A3,Def2;
  end;
  suppose
A4: x in REAL & y = +infty;
    then y - x = +infty by Def2;
    hence thesis by A4,Def2;
  end;
end;
