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 Th35:
  for c1,c2 being with_endpoints Curve of T
  st c1, c2 are_homotopic
  holds the_first_point_of c1 = the_first_point_of c2 &
  the_last_point_of c1 = the_last_point_of c2
  proof
    let c1,c2 be with_endpoints Curve of T;
    assume c1, c2 are_homotopic;
    then consider a,b be Point of T, p1,p2 be Path of a,b such that
A1: p1 = c1*L[01](0,1,inf dom c1,sup dom c1) &
    p2 = c2*L[01](0,1,inf dom c2,sup dom c2) &
    p1,p2 are_homotopic;
A2: 0 is Point of I[01] & 1 is Point of I[01] by BORSUK_1:40,XXREAL_1:1;
    consider f be Function of [:I[01],I[01]:], T such that
A3: f is continuous & for t being Point of I[01] holds f.(t,0) = p1.t &
    f.(t,1) = p2.t & f.(0,t) = a & f.(1,t) = b by A1;
A4:  f.(0,0)=p1.0 & f.(0,1)=p2.0 & f.(0,0)=a & f.(0,1)=a by A3,A2;
A5:  f.(1,0)=p1.1 & f.(1,1)=p2.1 & f.(1,0)=b & f.(1,1)=b by A3,A2;
A6: 0 in [.0,1.] & 1 in [.0,1.] by XXREAL_1:1;
A7:dom L[01](0,1,inf dom c1,sup dom c1)
    = the carrier of Closed-Interval-TSpace(0,1) by FUNCT_2:def 1
    .= [.0,1.] by TOPMETR:18;
A8:dom L[01](0,1,inf dom c2,sup dom c2)
    = the carrier of Closed-Interval-TSpace(0,1) by FUNCT_2:def 1
    .= [.0,1.] by TOPMETR:18;
A9:inf dom c1 <= sup dom c1 by XXREAL_2:40;
A10:inf dom c2 <= sup dom c2 by XXREAL_2:40;
A11:L[01](0,1,inf dom c2,sup dom c2).0
    = (sup dom c2 - inf dom c2)/(1 - 0) * (0 - 0) + inf dom c2
    by A10,BORSUK_6:35 .= inf dom c2;
    L[01](0,1,inf dom c1,sup dom c1).0
    = (sup dom c1 - inf dom c1)/(1 - 0) * (0 - 0) + inf dom c1
    by A9,BORSUK_6:35 .= inf dom c1;
    then p1.0 = c1.(inf dom c1) by A1,A6,A7,FUNCT_1:13;
    hence the_first_point_of c1 = the_first_point_of c2
    by A4,A1,A11,A6,A8,FUNCT_1:13;
A12:L[01](0,1,inf dom c2,sup dom c2).1
    = (sup dom c2 - inf dom c2)/(1 - 0) * (1 - 0) + inf dom c2
    by A10,BORSUK_6:35 .= sup dom c2;
    L[01](0,1,inf dom c1,sup dom c1).1
    = (sup dom c1 - inf dom c1)/(1 - 0) * (1 - 0) + inf dom c1
    by A9,BORSUK_6:35 .= sup dom c1;
    then p1.1 = c1.(sup dom c1) by A1,A6,A7,FUNCT_1:13;
    hence the_last_point_of c1 = the_last_point_of c2
    by A5,A1,A12,A6,A8,FUNCT_1:13;
  end;
