reserve T for non empty TopSpace,
  a, b, c, d for Point of T;
reserve X for non empty pathwise_connected TopSpace,
  a1, b1, c1, d1 for Point of X;

theorem Th77:
  for P, Q being Path of a, b st a, b are_connected & P, Q
  are_homotopic holds -P, -Q are_homotopic
proof
  reconsider fF = id I[01] as continuous Function of I[01], I[01];
  reconsider fB = L[01]((0,1)(#),(#)(0,1)) as continuous Function of I[01],
  I[01] by TOPMETR:20,TREAL_1:8;
  let P, Q be Path of a, b;
  assume
A1: a,b are_connected;
  set F = [:fB, fF:];
A2: dom fB = the carrier of I[01] by FUNCT_2:def 1;
  assume P, Q are_homotopic;
  then consider f being Function of [:I[01],I[01]:], T such that
A3: f is continuous and
A4: for s being Point of I[01] holds f.(s,0) = P.s & f.(s,1) = Q.s & for
  t being Point of I[01] holds f.(0,t) = a & f.(1,t) = b;
  reconsider ff = f * F as Function of [:I[01],I[01]:], T;
  take ff;
  thus ff is continuous by A3;
A5: 0 is Point of I[01] by BORSUK_1:43;
A6: for t being Point of I[01] holds ff.(0,t) = b & ff.(1,t) = a
  proof
A7: for t being Point of I[01], t9 being Real
     st t = t9 holds fB.t = 1-t9
    proof
      let t be Point of I[01], t9 be Real;
      assume
A8:   t = t9;
      reconsider ee = t as Point of Closed-Interval-TSpace (0,1) by TOPMETR:20;
A9:   (0,1)(#) = 1 & (#)(0,1) = 0 by TREAL_1:def 1,def 2;
      fB.t = fB.ee .= ((0-1)*t9 + 1) by A8,A9,TREAL_1:7
        .= (1 - 1*t9);
      hence thesis;
    end;
    then
A10: fB.0 = 1 - 0 by A5
      .= 1;
    1 is Point of I[01] by BORSUK_1:43;
    then
A11: fB.1 = 1 - 1 by A7
      .= 0;
    let t be Point of I[01];
A12: dom fF = the carrier of I[01];
    reconsider tt = t as Real;
A13: dom fB = the carrier of I[01] by FUNCT_2:def 1;
    then
A14: 0 in dom fB by BORSUK_1:43;
A15: dom F = [:dom fB, dom fF:] by FUNCT_3:def 8;
    then
A16: [0,t] in dom F by A14,ZFMISC_1:87;
A17: 1 in dom fB by A13,BORSUK_1:43;
    then
A18: [1,t] in dom F by A15,ZFMISC_1:87;
    F.(1,t) = [fB.1,fF.t] by A12,A17,FUNCT_3:def 8
      .= [0,tt] by A11,FUNCT_1:18;
    then
A19: ff.(1,t) = f.(0,t) by A18,FUNCT_1:13
      .= a by A4;
    F.(0,t) = [fB.0,fF.t] by A12,A14,FUNCT_3:def 8
      .= [1,tt] by A10,FUNCT_1:18;
    then ff.(0,t) = f.(1,t) by A16,FUNCT_1:13
      .= b by A4;
    hence thesis by A19;
  end;
  for s being Point of I[01] holds ff.(s,0) = (-P).s & ff.(s,1) = (-Q).s
  proof
    let s be Point of I[01];
A20: for t being Point of I[01], t9 being Real st t = t9 holds fB.t = 1-t9
    proof
      let t be Point of I[01], t9 be Real;
      assume
A21:  t = t9;
      reconsider ee = t as Point of Closed-Interval-TSpace (0,1) by TOPMETR:20;
A22:  (0,1)(#) = 1 & (#)(0,1) = 0 by TREAL_1:def 1,def 2;
      fB.t = fB.ee .= ((0-1)*t9 + 1) by A21,A22,TREAL_1:7
        .= (1 - 1*t9);
      hence thesis;
    end;
A23: fB.s = 1 - s by A20;
A24: 1 is Point of I[01] by BORSUK_1:43;
A25: dom F = [:dom fB, dom fF:] by FUNCT_3:def 8;
A26: 1 in dom fF by BORSUK_1:43;
    then
A27: [s,1] in dom F by A2,A25,ZFMISC_1:87;
A28: 0 in dom fF by BORSUK_1:43;
    then
A29: [s,0] in dom F by A2,A25,ZFMISC_1:87;
A30: 1-s is Point of I[01] by JORDAN5B:4;
    F.(s,1) = [fB.s,fF.1] by A2,A26,FUNCT_3:def 8
      .= [1-s,1] by A23,A24,FUNCT_1:18;
    then
A31: ff.(s,1) = f.(1-s,1) by A27,FUNCT_1:13
      .= Q.(1-s) by A4,A30
      .= (-Q).s by A1,BORSUK_2:def 6;
    F.(s,0) = [fB.s,fF.0] by A2,A28,FUNCT_3:def 8
      .= [1-s, 0] by A5,A23,FUNCT_1:18;
    then ff.(s,0) = f.(1-s,0) by A29,FUNCT_1:13
      .= P.(1-s) by A4,A30
      .= (-P).s by A1,BORSUK_2:def 6;
    hence thesis by A31;
  end;
  hence thesis by A6;
end;
