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 Th32:
  for r1,r2 being Real
  for t1,t2 being Point of T
  for p1 being Path of t1,t2
  st t1,t2 are_connected & r1 < r2
  holds p1*L[01](r1,r2,0,1) is with_endpoints Curve of T
  proof
    let r1,r2 be Real;
    let t1,t2 be Point of T;
    let p1 be Path of t1,t2;
    assume
A1: t1,t2 are_connected;
    assume
A2:r1 < r2;
    then
A3: L[01](r1,r2,0,1) is continuous
    Function of Closed-Interval-TSpace(r1,r2),Closed-Interval-TSpace(0,1)
    by BORSUK_6:34;
A4: p1 is continuous & p1.0 = t1 & p1.1 = t2 by A1,BORSUK_2:def 2;
    set c = p1*L[01](r1,r2,0,1);
    rng L[01](r1,r2,0,1) c= [#]Closed-Interval-TSpace(0,1) by RELAT_1:def 19;
    then rng L[01](r1,r2,0,1) c= dom p1 by FUNCT_2:def 1,TOPMETR:20;
    then dom c = dom L[01](r1,r2,0,1) by RELAT_1:27;
    then dom c = [#]Closed-Interval-TSpace(r1,r2) by FUNCT_2:def 1;
    then
A5:dom c = [.r1,r2.] by A2,TOPMETR:18;
A6:rng c c= [#]T;
    then reconsider c as PartFunc of R^1, T by A5,RELSET_1:4,TOPMETR:17;
    set S = R^1|dom c;
    dom c = [#]S by PRE_TOPC:def 5;
    then reconsider g = c as Function of S, T by A6,FUNCT_2:2;
A7: S = Closed-Interval-TSpace(r1,r2) by A2,A5,TOPMETR:19;
    reconsider p2 = p1 as Function of Closed-Interval-TSpace(0,1),T
    by TOPMETR:20;
    g is continuous by A4,A7,A3,TOPMETR:20,TOPS_2:46;
    then c is parametrized-curve by A5;
    then reconsider c as Curve of T by Th20;
    dom c is left_end & dom c is right_end by A2,A5,XXREAL_2:33;
    then c is with_first_point & c is with_last_point;
    hence thesis;
  end;
