
theorem
  for S1 being non empty feasible ManySortedSign for S2 being non empty
  Subsignature of S1 for S3 being non empty Subsignature of S2 for A being
  MSAlgebra over S1 holds A|S3 = (A|S2)|S3
proof
  let S1 be non empty feasible ManySortedSign;
  let S2 be non empty Subsignature of S1;
  let S3 be non empty Subsignature of S2;
  let A be MSAlgebra over S1;
  set f1 = id the carrier of S2, g1 = id the carrier' of S2;
  set f2 = id the carrier of S3, g2 = id the carrier' of S3;
  rng f2 = the carrier of S3;
  then
A1: f1*f2 = f2 by Th10,RELAT_1:53;
  rng g2 = the carrier' of S3;
  then
A2: g1*g2 = g2 by Th10,RELAT_1:53;
  f1,g1 form_morphism_between S2,S1 & f2,g2 form_morphism_between S3,S2 by Def2
;
  hence thesis by A1,A2,Th27;
end;
