
theorem Th6:
  for X being non empty TopSpace, I being non empty set, J being
TopStruct-yielding non-Empty ManySortedSet of I, f being Function of X, product
  J holds f is continuous iff for i being Element of I holds proj(J,i)*f is
  continuous
proof
  let X be non empty TopSpace, I be non empty set, J be TopStruct-yielding
  non-Empty ManySortedSet of I, f be Function of X, product J;
  set B = product_prebasis J;
  hereby
    assume
A1: f is continuous;
    let i be Element of I;
    proj(J,i) is continuous by Th5;
    hence proj(J,i)*f is continuous by A1,TOPS_2:46;
  end;
  assume
A2: for i being Element of I holds proj(J,i)*f is continuous;
A3: for P being Subset of product J st P in B holds f"P is open
  proof
    let P be Subset of product J;
    reconsider p = P as Subset of product Carrier J by Def3;
    assume P in B;
    then consider
    i being set, T being TopStruct, V being Subset of T such that
A4: i in I and
A5: V is open and
A6: T = J.i and
A7: p = product ((Carrier J) +* (i,V)) by Def2;
    reconsider j = i as Element of I by A4;
    reconsider V as Subset of J.j by A6;
    proj(J,j)*f is continuous & [#](J.j) <> {} by A2;
    then
A8: (proj(J,j)*f)"V is open by A5,A6,TOPS_2:43;
    proj(J,j)"V = p by A7,Th4;
    hence thesis by A8,RELAT_1:146;
  end;
  B is prebasis of product J by Def3;
  hence thesis by A3,YELLOW_9:36;
end;
