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 Th25:
  for t1,t2 being Point of T
  for p being Path of t1,t2 st t1,t2 are_connected
  holds p is with_endpoints Curve of T
  proof
    let t1,t2 be Point of T;
    let p be Path of t1,t2;
    assume t1,t2 are_connected;
    then reconsider c = p as Curve of T by Th22;
A1: [.0,1.] = dom c by BORSUK_1:40,FUNCT_2:def 1;
    0 in [.0,1.] by XXREAL_1:1;
    then inf dom c in dom c by A1,Th4;
    then dom c is left_end by XXREAL_2:def 5;
    then
A2: c is with_first_point;
    1 in [.0,1.] by XXREAL_1:1;
    then sup dom c in dom c by A1,Th4;
    then dom c is right_end by XXREAL_2:def 6;
    then c is with_last_point;
    hence thesis by A2;
  end;
