reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem Th68:
   r <> 0 implies omega-y (uReal.r * No_omega^ y) = Unique_No y
proof
  assume r<>0;
  then
A1: |. uReal.r * No_omega^ y .|, No_omega^ y are_commensurate &
  not uReal.r * No_omega^ y == 0_No by Th66,Th67;
  y == Unique_No y by SURREALO:def 10;
  then No_omega^ y == No_omega^ Unique_No y by Lm5;
  then |. uReal.r * No_omega^ y .|, No_omega^ Unique_No y are_commensurate
  by A1,Th5;
  hence thesis by A1,Def7;
end;
