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 Th39:
  C is connected & Component_of x c= C implies C = Component_of x
proof
  assume
A1: C is connected;
  consider F being Subset-Family of GX such that
A2: for A being Subset of GX holds (A in F iff A is connected & x in A) and
A3: Component_of x = union F by Def7;
  assume
A4: Component_of x c= C;
  x in Component_of x by Th38;
  then C in F by A1,A4,A2;
  then C c= Component_of x by A3,ZFMISC_1:74;
  hence thesis by A4;
end;
