
theorem
  for I being non empty set, J being TopStruct-yielding non-Empty
  ManySortedSet of I st for i being Element of I holds J.i is compact holds
  product J is compact
proof
  let I be non empty set, J be TopStruct-yielding non-Empty ManySortedSet of I;
  assume
A1: for i being Element of I holds J.i is compact;
  reconsider B=product_prebasis J as prebasis of product J by WAYBEL18:def 3;
  assume not product J is compact;
  then consider F being Subset of B such that
A2: [#](product J) c= union(F) and
A3: for G being finite Subset of F holds not [#](product J) c= union G by Th15;
  defpred P[set,Element of I] means for G being finite Subset of F holds not
  proj(J,$2)"({$1}) c= union G;
A4: for i being Element of I ex xi being Element of J.i st P[xi, i] by A1,A3
,Th22;
  consider f being Element of product J such that
A5: for i being Element of I holds P[f.i, i] from ElProductEx(A4);
  f in [#](product J);
  then consider A being set such that
A6: f in A and
A7: A in F by A2,TARSKI:def 4;
  reconsider G = {A} as finite Subset of F by A7,ZFMISC_1:31;
  consider i being Element of I, Ai being Subset of J.i such that
  Ai is open and
A8: proj(J,i)"Ai = A by A7,Th16;
  proj(J,i).f in Ai by A6,A8,FUNCT_1:def 7;
  then f.i in Ai by Th8;
  then {f.i} c= Ai by ZFMISC_1:31;
  then proj(J,i)"({f.i}) c= A by A8,RELAT_1:143;
  then proj(J,i)"({f.i}) c= union G by ZFMISC_1:25;
  hence contradiction by A5;
end;
