reserve a, r, s for Real;

theorem Th10:
  for S, T being non empty TopSpace, A, B being Subset of T, f
being Function of S,T st f is being_homeomorphism & A is_a_component_of B holds
  f"A is_a_component_of f"B
proof
  let S, T be non empty TopSpace, A, B being Subset of T, f be Function of S,T
  such that
A1: f is being_homeomorphism;
A2: rng f = [#]T by A1
    .= the carrier of T;
  set Y = f"A;
  given X being Subset of T|B such that
A3: X = A and
A4: X is a_component;
A5: the carrier of (T|B) = B by PRE_TOPC:8;
  then f"X c= f"B by RELAT_1:143;
  then reconsider Y as Subset of S|(f"B) by A3,PRE_TOPC:8;
  take Y;
  thus Y = f"A;
  X is connected by A4;
  then A is connected by A3,CONNSP_1:23;
  then f"A is connected by A1,TOPS_2:62;
  hence Y is connected by CONNSP_1:23;
  let Z being Subset of S|(f"B) such that
A6: Z is connected and
A7: Y c= Z;
A8: f.:Y c= f.:Z by A7,RELAT_1:123;
A9: f is one-to-one by A1;
A10: f is continuous by A1;
  the carrier of S|(f"B) = f"B by PRE_TOPC:8;
  then f.:Z c= f.:(f"B) by RELAT_1:123;
  then reconsider R = f.:Z as Subset of T|B by A5,A2,FUNCT_1:77;
  reconsider Z1 = Z as Subset of S by PRE_TOPC:11;
  dom f = the carrier of S by FUNCT_2:def 1;
  then
A11: Z1 c= dom f;
  Z1 is connected by A6,CONNSP_1:23;
  then f.:Z1 is connected by A10,TOPS_2:61;
  then
A12: R is connected by CONNSP_1:23;
  X = f.:Y by A3,A2,FUNCT_1:77;
  then X = R by A4,A12,A8;
  hence thesis by A3,A9,A11,FUNCT_1:94;
end;
