reserve T,T1,T2,S for non empty TopSpace;
reserve GY for non empty TopSpace,
  r,s for Real;

theorem Th10:
  for T being non empty TopSpace, a, b being Point of T, P being
  Path of a, b st a,b are_connected holds P, P are_homotopic
proof
  let T be non empty TopSpace;
  let a, b be Point of T;
  let P be Path of a, b;
  defpred Z[object, object] means $2 = P.($1`1);
A1: for x be object st x in [:the carrier of I[01], the carrier of I[01]:]
  ex y be object st y in the carrier of T & Z[x,y]
  proof
    let x be object;
    assume x in [:the carrier of I[01], the carrier of I[01]:];
    then x`1 in the carrier of I[01] by MCART_1:10;
    hence thesis by FUNCT_2:5;
  end;
  consider f being Function of [:the carrier of I[01], the carrier of I[01]:],
  the carrier of T such that
A2: for x being object st x in [:the carrier of I[01], the carrier of I[01]
  :] holds Z[x, f.x] from FUNCT_2:sch 1(A1);
  the carrier of [:I[01],I[01]:] = [:the carrier of I[01], the carrier of
  I[01]:] by BORSUK_1:def 2;
  then reconsider
  f as Function of the carrier of [:I[01],I[01]:], the carrier of T;
  reconsider f as Function of [:I[01],I[01]:], T;
  assume
A3: a,b are_connected;
A4: for t being Point of I[01] holds f.(0,t) = a & f.(1,t) = b
  proof
    let t be Point of I[01];
    set t0 = [0,t], t1 = [1,t];
    0 in the carrier of I[01] by Lm1;
    then t0 in [:the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
    then
A5: f.t0 = P.(t0`1) by A2;
    1 in the carrier of I[01] by Lm1;
    then t1 in [:the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
    then
A6: f.t1 = P.(t1`1) by A2;
    P.0 = a & P.1 = b by A3,Def2;
    hence thesis by A5,A6;
  end;
A7: for W being Point of [:I[01], I[01]:], G being a_neighborhood of f.W ex
  H being a_neighborhood of W st f.:H c= G
  proof
    let W be Point of [:I[01], I[01]:], G be a_neighborhood of f.W;
    W in the carrier of [:I[01], I[01]:];
    then
A8: W in [:the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def 2;
    then reconsider W1 = W`1 as Point of I[01] by MCART_1:10;
A9: ex x,y be object st [x,y] = W by A8,RELAT_1:def 1;
    reconsider G9 = G as a_neighborhood of P.W1 by A2,A8;
    the carrier of I[01] c= the carrier of I[01];
    then reconsider AI = the carrier of I[01] as Subset of I[01];
    AI = [#]I[01];
    then Int AI = the carrier of I[01] by TOPS_1:15;
    then
A10: W`2 in Int AI by A8,MCART_1:10;
    P is continuous by A3,Def2;
    then consider H be a_neighborhood of W1 such that
A11: P.:H c= G9;
    set HH = [:H, the carrier of I[01]:];
    HH c= [:the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:95;
    then reconsider HH as Subset of [:I[01], I[01]:] by BORSUK_1:def 2;
    W1 in Int H & Int HH = [:Int H, Int AI:] by BORSUK_1:7,CONNSP_2:def 1;
    then W in Int HH by A9,A10,ZFMISC_1:def 2;
    then reconsider HH as a_neighborhood of W by CONNSP_2:def 1;
    take HH;
    f.:HH c= G
    proof
      let a be object;
      assume a in f.:HH;
      then consider b be object such that
A12:  b in dom f and
A13:  b in HH and
A14:  a = f.b by FUNCT_1:def 6;
      reconsider b as Point of [:I[01], I[01]:] by A12;
A15:  dom P = the carrier of I[01] & b`1 in H by A13,FUNCT_2:def 1,MCART_1:10;
      dom f = [:the carrier of I[01], the carrier of I[01]:] by FUNCT_2:def 1;
      then f.b = P.(b`1) by A2,A12;
      then f.b in P.:H by A15,FUNCT_1:def 6;
      hence thesis by A11,A14;
    end;
    hence thesis;
  end;
  take f;
  for s being Point of I[01] holds f.(s,0) = P.s & f.(s,1) = P.s
  proof
    let s be Point of I[01];
    reconsider s0 = [s,0], s1 = [s,1] as set;
    1 in the carrier of I[01] by Lm1;
    then s1 in [:the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
    then
A16: Z[s1, f.s1] by A2;
    0 in the carrier of I[01] by Lm1;
    then s0 in [:the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
    then Z[s0, f.s0] by A2;
    hence thesis by A16;
  end;
  hence thesis by A7,A4;
end;
