reserve T,U for non empty TopSpace;
reserve t for Point of T;
reserve n for Nat;
reserve T for TopStruct;
reserve f for PartFunc of R^1, T;
reserve c for Curve of T;
reserve T for non empty TopStruct;

theorem Th34:
  for T being non empty TopStruct,
      c1,c2 being with_endpoints Curve of T,
      a,b being Point of T,
      p1,p2 being Path of a,b
  st c1 = p1 & c2 = p2 & a,b are_connected
  holds c1,c2 are_homotopic iff p1,p2 are_homotopic
  proof
    let T be non empty TopStruct;
    let c1,c2 be with_endpoints Curve of T;
    let a,b be Point of T;
    let p1,p2 be Path of a,b;
    assume
A1: c1 = p1 & c2 = p2;
    assume
A2: a,b are_connected;
A3: 0 is Point of I[01] & 1 is Point of I[01] by BORSUK_1:40,XXREAL_1:1;
A4: inf dom c1 = 0 & sup dom c1 = 1 & inf dom c2 = 0 & sup dom c2 = 1
    by A1,Th4;
A5: dom p1 = the carrier of I[01] &
    dom p2 = the carrier of I[01] by FUNCT_2:def 1;
    thus c1,c2 are_homotopic implies p1,p2 are_homotopic
    proof
      assume c1,c2 are_homotopic;
      then consider aa,bb be Point of T, pp1,pp2 be Path of aa,bb such that
A6:   pp1 = c1*L[01](0,1,inf dom c1,sup dom c1) &
      pp2 = c2*L[01](0,1,inf dom c2,sup dom c2) &
      pp1,pp2 are_homotopic;
      consider f be Function of [:I[01],I[01]:], T such that
A7:   f is continuous & for t being Point of I[01]
      holds f.(t,0) = pp1.t & f.(t,1) = pp2.t
      & f.(0,t) = aa & f.(1,t) = bb by A6;
A8:   pp1 = p1 & pp2 = p2 by A1,A6,A4,A5,Th1,RELAT_1:52,TOPMETR:20;
A9:  f.(0,0)=pp1.0 & f.(0,1)=pp2.0 & f.(0,0)=aa & f.(0,1)=aa by A7,A3;
A10:  f.(1,0)=pp1.1 & f.(1,1)=pp2.1 & f.(1,0)=bb & f.(1,1)=bb by A7,A3;
      aa = a & bb = b by A8,A9,A10,A2,BORSUK_2:def 2;
      hence p1,p2 are_homotopic by A7,A8;
    end;
    assume
A11: p1,p2 are_homotopic;
     p1 = p1*L[01](0,1,0,1) & p2 = p2*L[01](0,1,0,1)
     by A5,Th1,RELAT_1:52,TOPMETR:20;
     hence c1,c2 are_homotopic by A4,A1,A11;
  end;
