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

theorem Th16:
  for J being non empty non void Signature
  for S being J-extension Signature
  for T being MSAlgebra over J
  for Q being MSAlgebra over S st Q is T-extension
  for x st x in the carrier of J holds (the Sorts of T).x = (the Sorts of Q).x
  proof
    let J be non empty non void Signature;
    let S be J-extension Signature;
    J is Subsignature of S by Def2;
    then
A1: id the carrier of J, id the carrier' of J form_morphism_between J,S
    by INSTALG1:def 2;
    let T be MSAlgebra over J;
    let Q be MSAlgebra over S; assume
A2: Q|J = the MSAlgebra of T;
    Q|J = Q|(J, id the carrier of J, id the carrier' of J)
    by INSTALG1:def 4;
    then
A3: the Sorts of T = (the Sorts of Q)*(id the carrier of J) &
    dom id the carrier of J = the carrier of J by A1,A2,INSTALG1:def 3;
    let x; assume
A4: x in the carrier of J;
    hence (the Sorts of T).x = (the Sorts of Q).((id the carrier of J).x)
    by A3,FUNCT_1:13
    .= (the Sorts of Q).x by A4,FUNCT_1:17;
  end;
