
theorem
  for S1,S2,S3 being ManySortedSign
  st S1 is_rougher_than S2 & S2 is_rougher_than S3 holds S1 is_rougher_than S3
proof
  let S1,S2,S3 be ManySortedSign;
  given f1,g1 being Function such that
A1: f1,g1 form_morphism_between S2,S1 and
A2: rng f1 = the carrier of S1 and
A3: rng g1 = the carrier' of S1;
  given f2,g2 being Function such that
A4: f2,g2 form_morphism_between S3,S2 and
A5: rng f2 = the carrier of S2 and
A6: rng g2 = the carrier' of S2;
  take f = f1*f2, g = g1*g2;
  thus f,g form_morphism_between S3,S1 by A1,A4,Th28;
A7: dom f1 = the carrier of S2 by A1;
  dom g1 = the carrier' of S2 by A1;
  hence thesis by A2,A3,A5,A6,A7,RELAT_1:28;
end;
