reserve I for set;
reserve S for non empty non void ManySortedSign,
  U0, U1 for non-empty MSAlgebra over S;

theorem
  for U0 being feasible free MSAlgebra over S, A being free GeneratorSet
  of U0 for Z being MSSubset of U0 st Z c= A & GenMSAlg Z is feasible holds
  GenMSAlg Z is free
proof
  let U0 be feasible free MSAlgebra over S, A be free GeneratorSet of U0, Z be
  MSSubset of U0 such that
A1: Z c= A and
A2: GenMSAlg Z is feasible;
  reconsider Z1 = Z as MSSubset of GenMSAlg Z by MSUALG_2:def 17;
  the Sorts of GenMSAlg Z1 = the Sorts of GenMSAlg Z by EXTENS_1:18;
  then reconsider z = Z as GeneratorSet of GenMSAlg Z by MSAFREE:def 4;
  reconsider z1 = z as ManySortedSubset of A by A1,PBOOLE:def 18;
  take z;
  z is free
  proof
    reconsider D = the Sorts of GenMSAlg Z as MSSubset of U0 by MSUALG_2:def 9;
    let U1 be non-empty MSAlgebra over S, g be ManySortedFunction of z, the
    Sorts of U1;
    consider G being ManySortedFunction of A, the Sorts of U1 such that
A3: G||z1 = g by Th6;
    consider h being ManySortedFunction of U0, U1 such that
A4: h is_homomorphism U0, U1 and
A5: h || A = G by MSAFREE:def 5;
    reconsider H = h || D as ManySortedFunction of GenMSAlg Z, U1;
    take H;
    thus H is_homomorphism GenMSAlg Z, U1 by A2,A4,Th22;
    thus g = h || Z by A3,A5,Th5
      .= H || z by Th5;
  end;
  hence thesis;
end;
