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

theorem Th62:
  not x == 0_No & not (x+y) == 0_No &
    omega-y x = omega-y (x + y) &
    omega-r x = omega-r (x+y) implies |.y.| infinitely< |.x.|
proof
  set r = omega-r x, w = omega-y x, N = No_omega^ w, R = uReal.r;
  assume that
A1: not x == 0_No & not (x+y) == 0_No and
A2: w = omega-y (x + y) &
  r = omega-r (x+y);
  let s be positive Real;
A3:  |.x.|, N are_commensurate & |.x+y.|, N are_commensurate
  by A1,A2,Def7;
A4: 0_No <= uReal.s by SURREALI:def 8;
A5: |.x - N * R .| infinitely< |.x.| by A1,Def8;
  |.x +y - N * R .| infinitely< |.x+y.| by A1,A2,Def8;
  then
A6: |.x +y +- N * R .| infinitely< |.x.| by Th16,A3,Th4;
  N * R - N*R == 0_No by SURREALR:39;
  then
A7: - N * R +(N*R + -x) = (- N * R + N*R) + -x
  ==0_No +-x = -x by SURREALR:43,SURREALR:37;
A8: x -x ==0_No by SURREALR:39;
  x +y + -x = y +(x + -x) by SURREALR:37;
  then
A9: x +y + -x == y+0_No =y by A8,SURREALR:43;
  - - N*R = N*R;
  then - (x +- N * R) = N*R + -x by SURREALR:40;
  then x +y +- N * R +- (x +- N * R) = x +y + (- N * R +(N*R + -x))
  == x +y + -x by A7,SURREALR:43,SURREALR:37;
  then x +y +- N * R +- (x +- N * R) == y by A9,SURREALO:4;
  then |.y.| == |.x +y +- N * R + - (x +- N * R).| <=
  |.x +y +- N * R.| + |.- (x +- N * R).| by Th48,Th37;
  then |.y.| <= |.x +y +- N * R.| + |.- (x +- N * R).| by SURREALO:4;
  then |.y.| * uReal.s <= (|.x +y +- N * R.| + |.- (x +- N * R).|) * uReal.s
  == |.x +y +- N * R.| * uReal.s + |.- (x +- N * R).| * uReal.s
  by A4,SURREALR:67,75;
  then
A10: |.y.| * uReal.s <=
  |.x +y +- N * R.| * uReal.s + |.- (x +- N * R).| * uReal.s
  by SURREALO:4;
A11: |.x.| = |.x.| * uReal.1 by SURREALN:48;
  |.x +y +- N * R.| * uReal.(s *2)< |.x.| by A6;
  then
A12: |.x +y +- N * R.| * uReal.(s *2*(1/2)) <= |.x.| * uReal.(1* (1/2))
  by Th59,A11;
  |.x +- N * R.| * uReal.(s *2)< |.x.| by A5;
  then
A13: |.x +- N * R.| * uReal.(s *2*(1/2)) < |.x.| * uReal.(1* (1/2))
  by Th59,A11;
  |.x +- N * R.| * uReal.s == |. -(x +- N * R).| * uReal.s
  by SURREALR:51,Th40;
  then |.-(x +- N * R).| * uReal.(s *2*(1/2)) < |.x.| * uReal.(1* (1/2))
  by A13,SURREALO:4;
  then
A14: |.x +y +- N * R.| * uReal.s + |.- (x +- N * R).| * uReal.s
  < |.x.| * uReal.(1* (1/2)) + |.x.| * uReal.(1* (1/2)) by A12,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 +y +- N * R.| * uReal.s + |.- (x +- N * R).| * uReal.s < |.x.|
  by A14,SURREALO:4;
  hence thesis by A10,SURREALO:4;
end;
