
theorem
  for M being non empty set, L being complete continuous LATTICE holds
  Omega (M-TOP_prod (M => Sigma L)) = Sigma (M-POS_prod (M => L))
proof
  let M be non empty set, L be complete continuous LATTICE;
A1: the RelStr of Sigma L = the RelStr of L by YELLOW_9:def 4;
  reconsider S = Sigma L as injective T_0-TopSpace;
  Omega Sigma L = Sigma L by WAYBEL25:15;
  then
  the RelStr of Omega (M-TOP_prod (M => Sigma L)) = M-POS_prod (M => Sigma
  L) by WAYBEL25:14
    .= (Sigma L)|^M by YELLOW_1:def 5
    .= L|^M by A1,WAYBEL27:15;
  then Sigma (L|^M) = Sigma Omega (M-TOP_prod (M => S)) by Th16
    .= Omega (M-TOP_prod (M => Sigma L)) by Th15;
  hence thesis by YELLOW_1:def 5;
end;
