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

theorem Th15:
  for J being non empty non void Signature
  for S being J-extension Signature
  for T being MSAlgebra over J
  for Q1,Q2 being MSAlgebra over S
  st the MSAlgebra of Q1 = the MSAlgebra of Q2
  holds Q1 is T-extension implies Q2 is T-extension
  proof
    let J be non empty non void Signature;
    let S be J-extension Signature;
A1: J is Subsignature of S by Def2;
    let T be MSAlgebra over J;
    let Q1,Q2 be MSAlgebra over S; assume
A2: the MSAlgebra of Q1 = the MSAlgebra of Q2;
    assume
A3: Q1|J = the MSAlgebra of T;
    Q1|J = Q1|(J, id the carrier of J, id the carrier' of J) &
    Q2|J = Q2|(J, id the carrier of J, id the carrier' of J)
    by INSTALG1:def 4;
    hence Q2|J = the MSAlgebra of T by A3,A1,A2,INSTALG1:def 2,21;
  end;
