reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;
reserve S for non empty TopStruct,
  f for Function of T, S,
  H for Subset-Family of S;
reserve T for non empty TopSpace,
  S for TopSpace,
  P1 for Subset of S,
  f for Function of T, S;
reserve T for TopSpace,
  S for non empty TopSpace,
  P for Subset of T,
  f for Function of T, S;
reserve GX,GY for non empty TopSpace;

theorem
  for GX being non empty TopSpace st (for x,y being Point of GX ex GY st
(GY is connected & ex f being Function of GY,GX st f is continuous & x in rng(f
  )& y in rng(f))) holds GX is connected
proof
  let GX;
  assume
A1: for x,y being Point of GX ex GY st (GY is connected & ex f being
  Function of GY,GX st f is continuous & x in rng f & y in rng f);
  for A,B being Subset of GX st [#](GX) = A \/ B & A <> {}(GX) & B <> {}(
  GX) & A is open & B is open holds A meets B
  proof
    let A,B be Subset of GX;
    assume that
A2: [#](GX) = A \/ B and
A3: A <> {}(GX) and
A4: B <> {}(GX) and
A5: A is open & B is open;
    set v = the Element of B;
    set u = the Element of A;
    reconsider y=v as Point of GX by A2,A4,XBOOLE_0:def 3;
    reconsider x=u as Point of GX by A2,A3,XBOOLE_0:def 3;
    consider GY such that
A6: GY is connected and
A7: ex f being Function of GY,GX st f is continuous & x in rng f & y
    in rng f by A1;
    consider f being Function of GY,GX such that
A8: f is continuous and
A9: x in rng f and
A10: y in rng f by A7;
    f"([#] GX)=[#] GY by Th41;
    then
A11: f"A \/ f"B = [#] GY by A2,RELAT_1:140;
    rng f /\ B <> {} by A4,A10,XBOOLE_0:def 4;
    then rng f meets B by XBOOLE_0:def 7;
    then
A12: f"B <> {}(GY) by RELAT_1:138;
    rng f /\ A <> {} by A3,A9,XBOOLE_0:def 4;
    then rng f meets A by XBOOLE_0:def 7;
    then
A13: f"A <> {}(GY) by RELAT_1:138;
    [#]GX <> {};
    then f"A is open & f"B is open by A5,A8,Th43;
    then f"A meets f"B by A6,A11,A13,A12,CONNSP_1:11;
    then f"A /\ f"B <> {} by XBOOLE_0:def 7;
    then f"(A /\ B) <> {} by FUNCT_1:68;
    then rng f meets (A /\ B) by RELAT_1:138;
    then ex u1 being object st u1 in rng f & u1 in A /\ B by XBOOLE_0:3;
    hence thesis by XBOOLE_0:4;
  end;
  hence thesis by CONNSP_1:11;
end;
