
theorem Th18:
  for I being non empty set, J being TopStruct-yielding non-Empty
ManySortedSet of I, i being Element of I, Fi being non empty Subset-Family of J
  .i st [#](J.i) c= union(Fi) holds [#](product J) c= union the set of all
proj(J,i)"Ai where
  Ai is Element of Fi
proof
  let I be non empty set, J be TopStruct-yielding non-Empty ManySortedSet of I,
  i be Element of I, Fi be non empty Subset-Family of J.i;
  assume
A1: [#](J.i) c= union(Fi);
  let f be object;
  assume
A2: f in [#](product J);
  then reconsider f9=f as Element of product J;
  f9.i in [#](J.i);
  then consider Ai0 being set such that
A3: f9.i in Ai0 and
A4: Ai0 in Fi by A1,TARSKI:def 4;
  f9 in product Carrier J by A2,WAYBEL18:def 3;
  then f9 in dom proj(Carrier J,i) by CARD_3:def 16;
  then
A5: f9 in dom proj(J,i) by WAYBEL18:def 4;
  reconsider Ai0 as Element of Fi by A4;
  proj(J,i).f9 in Ai0 by A3,Th8;
  then proj (J,i)"Ai0 in the set of all proj(J,i)"Ai where Ai is Element of Fi
 & f9 in proj(J,i)"Ai0 by A5,FUNCT_1:def 7;
  hence thesis by TARSKI:def 4;
end;
