
theorem
  for S1 being feasible ManySortedSign for S2 being Subsignature of S1
  for S3 being Subsignature of S2 holds S3 is Subsignature of S1
proof
  let S1 be feasible ManySortedSign;
  let S2 be Subsignature of S1, S3 be Subsignature of S2;
  rng id the carrier of S3 = the carrier of S3;
  then
A1: (id the carrier of S2)*id the carrier of S3 = id the carrier of S3 by Th10,
RELAT_1:53;
  rng id the carrier' of S3 = the carrier' of S3;
  then
A2: (id the carrier' of S2)*id the carrier' of S3 = id the carrier' of S3 by
Th10,RELAT_1:53;
  id the carrier of S3,id the carrier' of S3 form_morphism_between S3,S2 &
id the carrier of S2,id the carrier' of S2 form_morphism_between S2,S1 by Def2;
  hence id the carrier of S3,id the carrier' of S3 form_morphism_between S3,
  S1 by A1,A2,PUA2MSS1:29;
end;
