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

theorem Th15:
  x,y are_commensurate & y infinitely< z implies
      x infinitely< z
proof
  assume
A1: x,y are_commensurate & y infinitely< z;
  then consider n be positive Nat such that
A2:x < y*uInt.n;
  let r be positive Real;
  0_No < uReal.r by SURREALI:def 8;
  then
A3: x *uReal.r < y*uInt.n * uReal.r by A2,SURREALR:70;
  uInt.n = uDyadic.n = uReal.n by SURREALN:46,def 5;
  then y*uInt.n * uReal.r == y * (uInt.n * uReal.r) ==
  y * uReal.(n*r) by SURREALR:69,SURREALR:51,SURREALN:57;
  then y*uInt.n * uReal.r == y * uReal.(n*r) by SURREALO:4;
  then x *uReal.r <= y * uReal.(n*r) by A3,SURREALO:4;
  hence thesis by A1,SURREALO:4;
end;
