reserve n for Element of NAT,
  a, b for Real;

theorem
  for X being non empty TopSpace, Y being non empty SubSpace of X, x1,
x2 being Point of X, y1, y2 being Point of Y, f being Path of y1,y2 st x1 = y1
  & x2 = y2 & y1,y2 are_connected holds f is Path of x1,x2
proof
  let X be non empty TopSpace, Y be non empty SubSpace of X, x1, x2 be Point
  of X, y1, y2 be Point of Y, f be Path of y1,y2 such that
A1: x1 = y1 & x2 = y2 and
A2: y1, y2 are_connected;
  the carrier of Y is Subset of X by TSEP_1:1;
  then reconsider g = f as Function of I[01], X by FUNCT_2:7;
  f is continuous by A2,BORSUK_2:def 2;
  then
A3: g is continuous by PRE_TOPC:26;
A4: g.0 = x1 & g.1 =x2 by A1,A2,BORSUK_2:def 2;
  then x1, x2 are_connected by A3;
  hence thesis by A4,A3,BORSUK_2:def 2;
end;
