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;

theorem
  for GY being TopSpace for F being Function of GX,GY st F is continuous
  & F.:[#]GX = [#]GY & GX is connected holds GY is connected
proof
  let GY be TopSpace;
  let F be Function of GX,GY such that
A1: F is continuous and
A2: F.:[#]GX = [#]GY and
A3: GX is connected;
  assume not GY is connected;
  then consider A, B being Subset of GY such that
A4: [#]GY = A \/ B and
A5: A <> {}GY and
A6: B <> {}GY and
A7: A is closed and
A8: B is closed and
A9: A misses B by Th10;
A10: F" A is closed by A1,A7,PRE_TOPC:def 6;
A11: F" B is closed by A1,A8,PRE_TOPC:def 6;
A12: A /\ B = {} by A9;
  F" A /\ F" B = F"(A /\ B) by FUNCT_1:68
    .= {} by A12;
  then
A13: F" A misses F" B;
 the carrier of GY is non empty by A5;
  then
A14: [#]GX = F"the carrier of GY by FUNCT_2:40
    .= F" A \/ F" B by A4,RELAT_1:140;
A15: rng F = F.:(the carrier of GX) by RELSET_1:22;
  then
A16: F"B <> {}GX by A2,A6,RELAT_1:139;
  F"A <> {}GX by A2,A5,A15,RELAT_1:139;
  hence contradiction by A3,A14,A13,A10,A11,A16,Th10;
end;
