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
  for c being with_endpoints Curve of T,
      t1,t2 being Point of T
  st c*L[01](0,1,inf dom c,sup dom c) is Path of t1,t2 & t1,t2 are_connected
  holds t1 = the_first_point_of c & t2 = the_last_point_of c
  proof
    let c be with_endpoints Curve of T;
    let t1,t2 be Point of T;
    assume
A1: c*L[01](0,1,inf dom c,sup dom c) is Path of t1,t2 & t1,t2 are_connected;
A2: inf dom c <= sup dom c by XXREAL_2:40;
A3: 0 in [.0,1.] & 1 in [.0,1.] by XXREAL_1:1;
A4: dom L[01](0,1,inf dom c,sup dom c)
    = the carrier of Closed-Interval-TSpace(0,1) by FUNCT_2:def 1
    .= [.0,1.] by TOPMETR:18;
A5: L[01](0,1,inf dom c,sup dom c).0
    = (sup dom c - inf dom c)/(1 - 0) * (0 - 0) + inf dom c
    by A2,BORSUK_6:35 .= inf dom c;
A6:L[01](0,1,inf dom c,sup dom c).1
    = (sup dom c - inf dom c)/(1 - 0) * (1 - 0) + inf dom c
    by A2,BORSUK_6:35 .= sup dom c;
    reconsider p = c*L[01](0,1,inf dom c,sup dom c) as Path of t1,t2 by A1;
A7: p.0 = the_first_point_of c by A5,A3,A4,FUNCT_1:13;
    p.1 = the_last_point_of c by A6,A3,A4,FUNCT_1:13;
    hence thesis by A7,A1,BORSUK_2:def 2;
  end;
