reserve U0,U1,U2,U3 for Universal_Algebra,
  n for Nat,
  x,y for set;
reserve A for non empty Subset of U0,
  o for operation of U0,
  x1,y1 for FinSequence of A;

theorem Th26:
  for U0 be with_const_op Universal_Algebra holds UniAlg_meet(U0)
  is commutative
proof
  let U0 be with_const_op Universal_Algebra;
  set o = UniAlg_meet(U0);
  for x,y be Element of Sub(U0) holds o.(x,y)=o.(y,x)
  proof
    let x,y be Element of Sub(U0);
    reconsider U1=x,U2=y as strict SubAlgebra of U0 by Def14;
A1: o.(x,y) = U1 /\ U2 & o.(y,x) = U2 /\ U1 by Def16;
A2: (the carrier of U1) meets (the carrier of U2) by Th17;
    then
    the carrier of(U2 /\ U1) = (the carrier of U2) /\ (the carrier of U1)
    & for B2 be non empty Subset of U0 st B2=the carrier of (U2/\U1) holds the
    charact of (U2/\U1) = Opers(U0,B2) & B2 is opers_closed by Def9;
    hence thesis by A1,A2,Def9;
  end;
  hence thesis by BINOP_1:def 2;
end;
