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

theorem Th11: :: MEMBER_1:4
  r+f = r+g implies f = g
proof
  assume
A1: r+f = r+g;
  per cases by XXREAL_0:14;
  suppose
A2: f in REAL;
    now
      assume not g in REAL;
      then g = +infty or g = -infty by XXREAL_0:14;
      hence contradiction by A1,A2;
    end;
    then
A3: ex c, d being Complex st r = c & g = d & r+g = c+d by Def2;
    ex a, b being Complex st r = a & f = b & r+f = a+ b by A2,Def2;
    hence thesis by A1,A3;
  end;
  suppose
A4: f = +infty;
    then r+f = +infty by Def2;
    hence thesis by A1,A4,Lm8;
  end;
  suppose
A5: f = -infty;
    then r+f = -infty by Def2;
    hence thesis by A1,A5,Lm9;
  end;
end;
