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 Th40:
  A is a_component iff ex x being Point of GX st A = Component_of x
proof
  hereby
    assume
A1: A is a_component;
    then A <> {}GX;
    then consider y being Point of GX such that
A2: y in A by PRE_TOPC:12;
    take x = y;
    consider F being Subset-Family of GX such that
A3: for B being Subset of GX holds B in F iff B is connected & x in B and
A4: union F = Component_of x by Def7;
    A in F by A1,A2,A3;
    then A c= union F by ZFMISC_1:74;
    hence A = Component_of x by A1,A4;
  end;
  given x being Point of GX such that
A5: A = Component_of x;
    for B being Subset of GX st B is connected holds A c= B implies A = B
    by A5,Th39;
    hence A is a_component by A5;
end;
