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 Th28:
  for c being with_endpoints Curve of T st dom c = [.0,1.]
  holds c is Path of the_first_point_of c,the_last_point_of c
  proof
    let c be with_endpoints Curve of T;
    assume
A1: dom c = [.0,1.];
    set t1 = the_first_point_of c,t2 = the_last_point_of c;
    reconsider f = c as parametrized-curve PartFunc of R^1,T
    by Th23;
    consider S be SubSpace of R^1, p be Function of S, T such that
A2: f = p & S = R^1|dom f & p is continuous by Def4;
    reconsider p as Function of I[01],T by A2,A1,BORSUK_1:40,FUNCT_2:def 1;
A3: S = I[01] by A2,A1,TOPMETR:19,20;
A4: p.0 = t1 by A1,A2,XXREAL_2:25;
A5: p.1 = t2 by A2,A1,XXREAL_2:29;
    then t1,t2 are_connected by A2,A3,A4;
    hence thesis by A3,A4,A5,A2,BORSUK_2:def 2;
  end;
