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;

theorem Th22:
  for t1,t2 being Point of T
  for p being Path of t1,t2 st t1,t2 are_connected holds p is Curve of T
  proof
    let t1,t2 be Point of T;
    let p be Path of t1,t2;
    assume t1,t2 are_connected;
    then
A1: p is continuous & p.0 = t1 & p.1 = t2 by BORSUK_2:def 2;
    per cases;
    suppose T is non empty;
      then
A2:   [#]I[01] = dom p by FUNCT_2:def 1;
      then
A3:  dom p c= [#]R^1 by PRE_TOPC:def 4;
      then reconsider A = dom p as Subset of R^1;
A4:  I[01] = R^1 | A by A2,BORSUK_1:40,TOPMETR:19,20;
      rng p c= [#]T;
      then reconsider c = p as PartFunc of R^1, T by A3,RELSET_1:4;
      reconsider c as parametrized-curve PartFunc of R^1, T
      by A2,A4,Def4,A1,BORSUK_1:40;
      c is Element of Curves T by Th20;
      hence thesis;
    end;
    suppose
A5:   T is empty;
      then reconsider c = p as PartFunc of R^1, T;
      c = {} by A5;
      then reconsider c as parametrized-curve PartFunc of R^1,T by Lm1;
      c is Element of Curves T by Th20;
      hence thesis;
    end;
  end;
