reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th69:
  U is a_component implies
  (TOP-REAL 2)|C`|U is pathwise_connected
proof
  set T = T2|C`;
  assume
A1: U is a_component;
  let a, b be Point of T|U;
A2: the carrier of T|U = U by PRE_TOPC:8;
A3: U <> {}T by A1,CONNSP_1:32;
  per cases;
  suppose
A4: a = b;
    reconsider TU = T|U as non empty TopSpace by A3;
    reconsider a as Point of TU;
    reconsider f = I[01] --> a as Function of I[01],T|U;
    take f;
    thus thesis by A4,BORSUK_1:def 14,def 15,TOPALG_3:4;
  end;
  suppose
A5: a <> b;
A6: T|U is SubSpace of T2 by TSEP_1:7;
    then reconsider a1 = a, b1 = b as Point of T2 by A3,PRE_TOPC:25;
    reconsider V = U as Subset of T2 by PRE_TOPC:11;
    V is_a_component_of C` by A1;
    then
A7: V is open by SPRECT_3:8;
    U is connected by A1;
    then V is connected by CONNSP_1:23;
    then consider P being Subset of T2 such that
A8: P is_S-P_arc_joining a1,b1 and
A9: P c= V by A2,A3,A5,A7,TOPREAL4:29;
A10: a1 in P by A8,TOPREAL4:3;
    P is_an_arc_of a1,b1 by A8,TOPREAL4:2;
    then consider g being Function of I[01], T2|P such that
A11: g is being_homeomorphism and
A12: g.0 = a and
A13: g.1 = b by TOPREAL1:def 1;
A14: the carrier of T2|P = P by PRE_TOPC:8;
    then reconsider f = g as Function of I[01], T|U by A2,A9,A10,FUNCT_2:7;
    take f;
    T2|P is SubSpace of T|U by A2,A6,A9,A14,TSEP_1:4;
    hence f is continuous by A11,PRE_TOPC:26;
    thus thesis by A12,A13;
  end;
end;
