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

theorem Th13:
  for r be positive Real, x,y be Surreal st x infinitely< y
     holds x * uReal.r infinitely< y &
            x infinitely< y * uReal.r
proof
  let r be positive Real, x,y be Surreal such that
A1: x infinitely< y;
  thus x * uReal.r infinitely< y
  proof
    let s be positive Real;
    (x * uReal.r) * uReal.s == x * (uReal.s * uReal.r) ==
    x * uReal.(s* r) by SURREALN:57,SURREALR:69,SURREALR:51;
    then (x * uReal.r) * uReal.s == x * uReal.(s * r) < y
    by A1,SURREALO:4;
    hence thesis by SURREALO:4;
  end;
  let s be positive Real;
A2: 0_No < uReal.r by SURREALI:def 8;
A3: x * uReal.(s*(1/r))*uReal.r < y * uReal.r by A1,A2,SURREALR:70;
  s*((1/r)*r) =s * 1 by XCMPLX_1:106;
  then s*(1/r)*r =s;
  then x * uReal.(s*(1/r))*uReal.r == x * (uReal.(s*(1/r))*uReal.r) ==
  x* uReal.s by SURREALR:69,SURREALR:51, SURREALN:57;
  then x * uReal.(s*(1/r))*uReal.r == x* uReal.s by SURREALO:4;
  hence x * uReal.s < y * uReal.r by A3,SURREALO:4;
end;
