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 Th26:
  for c being Curve of T
  for r1,r2 being Real
  holds c | [.r1, r2.] is Curve of T
  proof
    let c be Curve of T;
    let r1,r2 be Real;
    reconsider f = c as parametrized-curve PartFunc of R^1, T
    by Th23;
    set f1 = f | [.r1, r2.];
    reconsider A = dom f as interval Subset of REAL by Def4;
    reconsider B = [.r1,r2.] as interval Subset of REAL;
A1: A /\ B is interval Subset of REAL;
    then
A2: dom f1 is interval Subset of REAL by RELAT_1:61;
    consider S be SubSpace of R^1, g be Function of S,T such that
A3: f = g & S = R^1|dom f & g is continuous by Def4;
    reconsider A0 = dom f as Subset of R^1;
A4: [#]S = A0 by A3,PRE_TOPC:def 5;
    reconsider K0 = (dom f)/\[.r1,r2.] as Subset of S by A4,XBOOLE_1:17;
    reconsider g1 = g|K0 as Function of S|K0,T by PRE_TOPC:9;
A5: g1 is continuous by A3,TOPMETR:7;
A6: g1 = (f|dom f) | [.r1,r2.] by A3,RELAT_1:71 .= f1;
A7:(dom f)/\[.r1,r2.] = dom f1 by RELAT_1:61;
    reconsider K1 = K0 as Subset of R^1|A0 by A3;
    reconsider D1 = dom f1 as Subset of R^1 by A1,RELAT_1:61,TOPMETR:17;
    S|K0 = R^1 | D1 by A3,A7,PRE_TOPC:7,XBOOLE_1:17;
    then reconsider f1 as parametrized-curve PartFunc of R^1, T
    by A2,A5,A6,Def4;
    c | [.r1, r2.] = f1;
    hence thesis by Th20;
  end;
