
theorem Th27:
  for S1,S2,S3 being non empty ManySortedSign for f1,g1 being
Function st f1,g1 form_morphism_between S1,S2 for f2,g2 being Function st f2,g2
form_morphism_between S2,S3 for A being MSAlgebra over S3 holds A|(S1,f2*f1,g2*
  g1) = (A|(S2,f2,g2))|(S1,f1,g1)
proof
  let S1,S2,S3 be non empty ManySortedSign;
  let f1,g1 be Function such that
A1: f1,g1 form_morphism_between S1,S2;
  let f2,g2 be Function such that
A2: f2,g2 form_morphism_between S2,S3;
A3: f2*f1, g2*g1 form_morphism_between S1,S3 by A1,A2,PUA2MSS1:29;
  let A be MSAlgebra over S3;
A4: the Charact of (A|(S2,f2,g2))|(S1,f1,g1) = (the Charact of A|(S2,f2,g2))
  *g1 by A1,Def3
    .= (the Charact of A)*g2*g1 by A2,Def3
    .= (the Charact of A)*(g2*g1) by RELAT_1:36;
  the Sorts of (A|(S2,f2,g2))|(S1,f1,g1) = (the Sorts of A|(S2,f2,g2))*f1
  by A1,Def3
    .= (the Sorts of A)*f2*f1 by A2,Def3
    .= (the Sorts of A)*(f2*f1) by RELAT_1:36;
  hence thesis by A3,A4,Def3;
end;
