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

theorem
  x+y = 0 implies x = -y
proof
  assume
A1: x+y = 0;
  per cases by XXREAL_0:14;
  suppose
    x in REAL;
    then reconsider x as Real;
    x+y is real by A1;
    then reconsider y as Real;
    x+y = 0 by A1;
    then x = -y;
    hence thesis;
  end;
  suppose
A2: x = -infty;
    then y = +infty by A1,Def2;
    hence thesis by A2,Def3;
  end;
  suppose
A3: x = +infty;
    then y = -infty by A1,Def2;
    hence thesis by A3,Def3;
  end;
end;
