reserve X,Y for set, x,y,z for object, i,j,n for natural number;

theorem Th22:
  for J being non void Signature
  for S being J-extension non void Signature
  for A1,A2 being MSAlgebra over S st the MSAlgebra of A1 = the MSAlgebra of A2
  holds A1|J = A2|J
  proof
    let J be non void Signature;
    let S be J-extension non void Signature;
    let A1,A2 be MSAlgebra over S such that
A1: the MSAlgebra of A1 = the MSAlgebra of A2;
A2: A1|J = A1|(J, id the carrier of J, id the carrier' of J) &
    A2|J = A2|(J, id the carrier of J, id the carrier' of J) by INSTALG1:def 4;
    J is Subsignature of S by Def2;
    then
    id the carrier of J, id the carrier' of J form_morphism_between J, S
    by INSTALG1:def 2;
    then the Sorts of A1|J = (the Sorts of A1)*id the carrier of J &
    the Sorts of A2|J = (the Sorts of A2)*id the carrier of J &
    the Charact of A1|J = (the Charact of A1)*id the carrier' of J &
    the Charact of A2|J = (the Charact of A2)*id the carrier' of J
    by A2,INSTALG1:def 3;
    hence A1|J = A2|J by A1;
  end;
