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

theorem Th63:
  |.y.| infinitely< |.x.| implies
      not x == 0_No & not (x+y) == 0_No &
       omega-y x = omega-y (x + y) &
       omega-r x = omega-r (x+y)
proof
  assume
A1: |.y.| infinitely< |.x.|;
  hence
A2: not x == 0_No & not (x+y) == 0_No by Th45,Th47;
A3: |.y.| * uReal.((2+1)/2) < |.x.| by A1;
  then |.x.|,|.x+y.| are_commensurate by Th57;
  hence
A4: omega-y x = omega-y (x + y) by A2,Th61;
  set N = No_omega^ omega-y x;
A5:  |.x - N * uReal.omega-r x .| infinitely< |.x.| by A2,Def8;
  |.x +y +- N * uReal.omega-r x .| infinitely< |.x.|
  proof
    let s be positive Real;
A6: 0_No <= uReal.s by SURREALI:def 8;
    set r = uReal.omega-r x;
    x +y +- N * r = y +(x +- N * r) by SURREALR:37;
    then |.x +y +- N * r .|*uReal.s <= (|.x +- N * r .| +|.y.|)*uReal.s
    == |.x +- N * r .|*uReal.s + |.y.| *uReal.s by Th37,A6,SURREALR:67,75;
    then
A7: |.x +y +- N * r .|*uReal.s <=
    |.x +- N * r .|*uReal.s + |.y.| *uReal.s by SURREALO:4;
A8: |.x.| = |.x.| * uReal.1 by SURREALN:48;
    |.x +- N * r.| * uReal.(s *2)< |.x.| by A5;
    then
A9: |.x +- N * r.| * uReal.(s *2*(1/2)) <= |.x.| * uReal.(1* (1/2))by Th59,A8;
    |. y .| * uReal.(s *2)< |.x.| by A1;
    then |.y.| * uReal.(s *2*(1/2)) < |.x.| * uReal.(1* (1/2)) by Th59,A8;
    then
A10: |.x +- N * r.| * uReal.s + |. y .| * uReal.s
    < |.x.| * uReal.(1* (1/2)) + |.x.| * uReal.(1* (1/2))
    by A9,SURREALR:44;
    |.x.| * uReal.(1* (1/2)) + |.x.| * uReal.(1* (1/2))
    == |.x.| *(uReal.(1/2)+uReal.(1/2)) == |.x.| *(uReal.((1/2)+(1/2)))
    by SURREALR:67,51,SURREALN:55;
    then |.x.| * uReal.(1* (1/2)) + |.x.| * uReal.(1* (1/2)) ==
    |.x.| *(uReal.1) = |.x.| by SURREALN:48,SURREALO:4;
    then |.x +- N * r.| * uReal.s + |. y .| * uReal.s < |.x.|
    by A10,SURREALO:4;
    hence thesis by A7,SURREALO:4;
  end;
  then |.x +y - N * uReal.omega-r x .| infinitely< |.x+y.| by A3,Th57,Th16;
  hence thesis by A2,A4,Def8;
end;
