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

theorem Th12:
  for U0 being with_const_op Universal_Algebra for H be non empty
  Subset of Sub(U0) holds meet ((Carr U0).:H) is non empty Subset of U0
proof
  let U0 be with_const_op Universal_Algebra;
  let H be non empty Subset of Sub(U0);
  set u = the Element of Constants(U0);
  reconsider CH = (Carr U0).:H as Subset-Family of U0;
A1: for S being set st S in (Carr U0).:H holds u in S
  proof
    let S be set;
    assume
A2: S in (Carr U0).:H;
    then reconsider S as Subset of U0;
    consider X1 being Element of Sub(U0) such that
    X1 in H and
A3: S = (Carr U0).X1 by A2,FUNCT_2:65;
    reconsider X1 as strict SubAlgebra of U0 by UNIALG_2:def 14;
    S = the carrier of X1 by A3,Def4;
    hence thesis by Th11;
  end;
  CH <> {} by Th9;
  hence thesis by A1,SETFAM_1:def 1;
end;
