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 Th33:
  for c being with_endpoints Curve of T
  holds the_first_point_of c, the_last_point_of c are_connected
  proof
    let c be with_endpoints Curve of T;
    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, g be Function of S, T such that
A1: f = g & S = R^1|dom f & g is continuous by Def4;
    set r1 = inf dom c, r2 = sup dom c;
    set p = g*L[01](0,1,r1,r2);
A2: r1 <= r2 by XXREAL_2:40;
    then
A3: L[01](0,1,r1,r2) is continuous
    Function of Closed-Interval-TSpace(0,1),Closed-Interval-TSpace(r1,r2)
    by BORSUK_6:34;
    rng L[01](0,1,r1,r2) c= [#]Closed-Interval-TSpace(r1,r2) by RELAT_1:def 19;
    then rng L[01](0,1,r1,r2) c= [.r1,r2.] by A2,TOPMETR:18;
    then rng L[01](0,1,r1,r2) c= dom c by Th27;
    then dom p = dom L[01](0,1,r1,r2) by A1,RELAT_1:27;
    then
A4:dom p = [#]Closed-Interval-TSpace(0,1) by FUNCT_2:def 1;
    rng p c= [#]T;
    then reconsider p as Function of I[01],T by A4,FUNCT_2:2,TOPMETR:20;
    dom f = [.r1,r2.] by Th27;
    then S = Closed-Interval-TSpace(r1,r2) by A1,A2,TOPMETR:19;
    then
A5: p is continuous by A1,A3,TOPMETR:20,TOPS_2:46;
    dom p = [.0,1.] by A4,TOPMETR:18;
    then
A6: 0 in dom p & 1 in dom p by XXREAL_1:1;
A7: L[01](0,1,r1,r2).0
    = ((r2 - r1)/(1 - 0)) * (0  - 0) + r1 by A2,BORSUK_6:35 .= r1;
A8: L[01](0,1,r1,r2).1
    = ((r2 - r1)/(1 - 0)) * (1  - 0) + r1 by A2,BORSUK_6:35 .= r2;
A9: p.0
    = t1 by A1,A7,A6,FUNCT_1:12;
    p.1
    = t2 by A1,A8,A6,FUNCT_1:12;
    hence thesis by A5,A9;
  end;
