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

theorem Th59:
  x*uReal.r1 < y*uReal.r2 & 0 < r implies x*uReal.(r1*r) < y*uReal.(r2 *r)
proof
  assume
A1:x*uReal.r1 < y*uReal.r2 & 0 < r;
  then 0_No < uReal.r by SURREALI:def 8;
  then
A2: x*uReal.r1*uReal.r < y*uReal.r2 *uReal.r by A1,SURREALR:70;
  x*uReal.r1*uReal.r == x*uReal.(r1*r) by Lm1;
  then x*uReal.(r1*r) < y*uReal.r2 *uReal.r ==
  y*uReal.(r2 *r) by A2,SURREALO:4,Lm1;
  hence thesis by SURREALO:4;
end;
