reserve GX for TopSpace;
reserve A, B, C for Subset of GX;
reserve TS for TopStruct;
reserve K, K1, L, L1 for Subset of TS;
reserve GX for non empty TopSpace;
reserve A, C for Subset of GX;
reserve x for Point of GX;

theorem
  for F being Subset-Family of GX st
  for A being Subset of GX holds A in F iff A is a_component holds
  F is Cover of GX
proof
  let F be Subset-Family of GX such that
A1: for A being Subset of GX holds A in F iff A is a_component;
  now
    let x be object;
    assume x in [#]GX;
    then reconsider y = x as Point of GX;
    Component_of y is a_component by Th40;
    then Component_of y in F by A1;
    then
A2: Component_of y c= union F by ZFMISC_1:74;
    y in Component_of y by Th38;
    hence x in union F by A2;
  end;
  then [#]GX c= union F by TARSKI:def 3;
  hence thesis by SETFAM_1:def 11;
end;
