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

theorem Th16:
  x,y are_commensurate & z infinitely< x implies
      z infinitely< y
proof
  assume
A1: x,y are_commensurate & z infinitely< x;
  then consider n be positive Nat such that
A2:x < y*uInt.n;
  let r be positive Real;
  z *uReal.(r*n) <= x by A1;
  then
A3:z *uReal.(r*n) < y*uInt.n by SURREALO:4,A2;
A4: 0_No < uReal.(1/n) by SURREALI:def 8;
  uInt.n = uDyadic.n = uReal.n by SURREALN:46,def 5;
  then z *uReal.(r*n)*uReal.(1/n) < y*uReal.n * uReal.(1/n)==y
  by A4,Lm2,A3,SURREALR:70;
  then
A5:z *uReal.(r*n)*uReal.(1/n) < y by SURREALO:4;
  r*n*(1/n) = r*(n*(1/n))= r*1 by XCMPLX_1:106;
  then z *uReal.(r*n)*uReal.(1/n) == z *(uReal.(r*n)*uReal.(1/n)) ==
  z * uReal.r by SURREALR:69,SURREALR:51,SURREALN:57;
  then z *uReal.(r*n)*uReal.(1/n) == z * uReal.r by SURREALO:4;
  hence thesis by A5,SURREALO:4;
end;
