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

theorem Th5:
  for x,y,z be Surreal st
    x==y & x,z are_commensurate holds y,z are_commensurate
proof
  let x,y,z be Surreal such that
A1:x==y & x,z are_commensurate;
  thus ex n be positive Nat st y < uInt.n * z
  proof
    consider n be positive Nat such that
A2: x < uInt.n * z by A1;
    y < uInt.n * z by A1,A2,SURREALO:4;
    hence thesis;
  end;
  consider n be positive Nat such that
A3: z < uInt.n * x by A1;
  uInt.n * x == uInt.n * y by A1,SURREALR:51;
  then z < uInt.n * y by A3,SURREALO:4;
  hence thesis;
end;
