
theorem Th5:
  for I being non empty set, J being TopStruct-yielding non-Empty
  ManySortedSet of I, i being Element of I holds proj(J,i) is continuous
proof
  let I be non empty set, J be TopStruct-yielding non-Empty ManySortedSet of I,
  i be Element of I;
A1: for P being Subset of J.i st P is open holds proj(J,i)"P is open
  proof
    let P be Subset of J.i;
    assume
A2: P is open;
    proj(J,i)"P c= product Carrier J
    proof
      let x be object;
      assume x in proj(J,i)"P;
      then x in dom proj(Carrier J,i) by FUNCT_1:def 7;
      hence thesis;
    end;
    then reconsider x = proj(J,i)"P as Subset of product Carrier J;
    product_prebasis J is prebasis of product J by Def3;
    then
A3: product_prebasis J c= the topology of product J by TOPS_2:64;
    x = product ((Carrier J) +* (i,P)) by Th4;
    then proj(J,i)"P in product_prebasis J by A2,Def2;
    hence thesis by A3;
  end;
  [#](J.i) <> {};
  hence thesis by A1,TOPS_2:43;
end;
