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

theorem Th25:
  -(x + y) = -y - x
proof
  per cases by XXREAL_0:14;
  suppose
A1: x = +infty & y = +infty;
    hence -(x + y) = -+infty by Def2
      .= -y - x by A1,Def2,Lm10;
  end;
  suppose
    x = +infty & y = -infty;
    hence thesis by Lm4;
  end;
  suppose
    x = +infty & y in REAL;
    hence thesis by Lm6;
  end;
  suppose
    x = -infty & y = +infty;
    hence thesis by Lm4;
  end;
  suppose
A2: x = -infty & y = -infty;
    hence -(x + y) = --infty by Def2
      .= -y - x by A2,Def2,Th5;
  end;
  suppose
    x = -infty & y in REAL;
    hence thesis by Lm11;
  end;
  suppose
    x in REAL & y = +infty;
    hence thesis by Lm6;
  end;
  suppose
    x in REAL & y = -infty;
    hence thesis by Lm11;
  end;
  suppose
    x in REAL & y in REAL;
    then reconsider a=x, b=y as Real;
    -(x + y) = -(a + b) .= -a + - b;
    hence thesis;
  end;
end;
