reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;
reserve n for Nat;

theorem Th41:
  x==y implies Unique_No x = Unique_No y
proof
  assume A1: x==y;
  then A2:born_eq x = born_eq y by Th33;
  Unique_No y == y by Def10;
  then Unique_No y == x & Unique_No y in (unique_No_op born_eq x).born_eq x
  by A2,Def10,A1,Th4;
  hence thesis by Def10;
end;
