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 Th29:
  for c being with_endpoints Curve of T holds
  c*L[01](0,1,inf dom c,sup dom c)
  is Path of the_first_point_of c,the_last_point_of c
  proof
    let c be with_endpoints Curve of T;
    set t1 = the_first_point_of c, t2 = the_last_point_of c;
    reconsider c0 = c as parametrized-curve PartFunc of R^1, T
    by Th23;
    consider S be SubSpace of R^1, g be Function of S, T such that
A1:c0 = g & S = R^1|dom c0 & g is continuous by Def4;
    reconsider S as non empty TopStruct by A1;
A2:inf dom c <= sup dom c by XXREAL_2:40;
    then
A3:L[01](0,1,inf dom c,sup dom c) is continuous
    Function of Closed-Interval-TSpace(0,1),
    Closed-Interval-TSpace(inf dom c,sup dom c) by BORSUK_6:34;
A4:dom c0 = [.inf dom c,sup dom c.] by Th27;
    then
A5:Closed-Interval-TSpace(inf dom c,sup dom c) = S by A2,A1,TOPMETR:19;
    reconsider f = L[01](0,1,inf dom c,sup dom c) as Function of I[01],S
    by A4,A2,A1,TOPMETR:19,20;
    reconsider p = g*f as Function of I[01],T;
A6: 0 in [.0,1.] & 1 in [.0,1.] by XXREAL_1:1;
A7: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;
A8: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;
A9: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;
A10: p is continuous by A1,A3,A5,TOPMETR:20,TOPS_2:46;
A11: p.0 = t1 by A8,A1,A6,A7,FUNCT_1:13;
A12: p.1 = t2 by A9,A1,A6,A7,FUNCT_1:13;
    then t1,t2 are_connected by A10,A11;
    hence thesis by A1,A10,A11,A12,BORSUK_2:def 2;
  end;
