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 Th38:
  x in Component_of x
proof
  consider F being Subset-Family of GX such that
A1: for A being Subset of GX holds A in F iff A is connected & x in A and
A2: Component_of x = union F by Def7;
A3: for A being set holds A in F implies x in A by A1;
  F <> {} by A1,Th31;
  then
A4: x in meet F by A3,SETFAM_1:def 1;
  meet F c= union F by SETFAM_1:2;
  hence thesis by A2,A4;
end;
