theorem Th50:
  X (+) Y = (X (o) Y) (+) Y & X (-) Y = (X (O) Y) (-) Y
proof
  (X (o) Y) (+) Y =(X (+) Y) (O) Y;
  then
A1: (X (o) Y) (+) Y c= X (+) Y by Th41;
  X c= X (o) Y by Th41;
  then X (+) Y c= (X (o) Y) (+) Y by Th9;
  hence X (+) Y = (X (o) Y) (+) Y by A1;
  (X (O) Y) (-) Y = (X (-) Y) (o) Y;
  hence X (-) Y c= (X (O) Y) (-) Y by Th41;
  X (O) Y c= X by Th41;
  hence thesis by Th9;
end;
