reserve U0 for Universal_Algebra,
  U1 for SubAlgebra of U0,
  o for operation of U0;

theorem Th14:
  for U0 being with_const_op Universal_Algebra for H be non empty
Subset of Sub(U0) for S being non empty Subset of U0 st S = meet ((Carr U0).:H)
  holds S is opers_closed
proof
  let U0 be with_const_op Universal_Algebra;
  let H be non empty Subset of Sub(U0);
  let S be non empty Subset of U0 such that
A1: S = meet ((Carr U0).:H);
A2: (Carr U0).:H <> {} by Th9;
  for o be operation of U0 holds S is_closed_on o
  proof
    let o be operation of U0;
    let s be FinSequence of S;
    assume
A3: len s = arity o;
    now
      let a be set;
      assume
A4:   a in (Carr U0).:H;
      then reconsider H1 = a as Subset of U0;
      consider H2 being Element of Sub U0 such that
      H2 in H and
A5:   H1 = (Carr U0).H2 by A4,FUNCT_2:65;
A6:   H1 = the carrier of H2 by A5,Def4;
      then reconsider H3 = H1 as non empty Subset of U0;
      S c= H1 by A1,A4,SETFAM_1:3;
      then reconsider s1 = s as FinSequence of H3 by FINSEQ_2:24;
      H3 is opers_closed by A6,UNIALG_2:def 7;
      then H3 is_closed_on o;
      then o.s1 in H3 by A3;
      hence o.s in a;
    end;
    hence thesis by A1,A2,SETFAM_1:def 1;
  end;
  hence thesis;
end;
