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

theorem Th53:
  0_No <= x infinitely< y implies |.x*uReal.r.| infinitely< y
proof
  assume
A1:0_No <= x infinitely< y;
  let s be positive Real;
  per cases;
  suppose 0_No <= x*uReal.r;
    then |.x*uReal.r.| = x*uReal.r by Def6;
    then |.x*uReal.r.| * uReal.s == x*uReal.(r *s) < y by Th20,A1,Lm1;
    hence thesis by SURREALO:4;
  end;
  suppose x*uReal.r < 0_No;
    then
A2: |.x*uReal.r.| = - (x*uReal.r) by Def6;
    x* (- uReal.r) == x*uReal.-r by SURREALN:56,SURREALR:51;
    then - (x*uReal.r) == x*uReal.-r by SURREALR:58;
    then |.x*uReal.r.| * uReal.s == x*uReal.(-r) *uReal.s ==
    x*uReal.((-r) *s) by A2,SURREALR:51,Lm1;
    then |.x*uReal.r.| * uReal.s == x*uReal.((-r) *s) < y
    by Th20,A1,SURREALO:4;
    hence thesis by SURREALO:4;
  end;
end;
