
theorem Th58:
  for I being non empty set
  for J being TopSpace-yielding non-Empty ManySortedSet of I
  for P being non empty Subset of product Carrier J st P in product_prebasis J
  holds (for j being Element of I holds proj(J,j).:P is open) &
    ex i being Element of I
    st for j being Element of I st j <> i holds proj(J,j).:P = [#](J.j)
proof
  let I being non empty set;
  let J being TopSpace-yielding non-Empty ManySortedSet of I;
  let P be non empty Subset of product Carrier J;
  assume P in product_prebasis J;
  then consider i being Element of I such that
    A1: proj(J,i).:P is open and
    A2: for j being Element of I st j <> i holds proj(J,j).:P = [#](J.j)
    by Th57;
  hereby
    let j be Element of I;
    j<>i implies proj(J,j).:P = [#](J.j) by A2;
    hence proj(J,j).:P is open by A1;
  end;
  take i;
  thus thesis by A2;
end;
