
theorem Th14:
  for T being non empty TopSpace, A,B being Subset of T, G being
  Rain of A,B, p being Point of T, S being non empty Subset of ExtREAL st S =
Rainbow(p,G) holds for e1 being R_eal st e1 = 1 holds 0. <= sup S & sup S <= e1
proof
  reconsider a = 0,b = 1 as R_eal by XXREAL_0:def 1;
  let T be non empty TopSpace, A,B be Subset of T, G be Rain of A,B, p be
  Point of T, S be non empty Subset of ExtREAL;
  consider s being object such that
A1: s in S by XBOOLE_0:def 1;
  reconsider s as R_eal by A1;
  assume S = Rainbow(p,G);
  then S c= DYADIC by Th13;
  then
A2: S c= [.a,b.] by URYSOHN2:28;
  let e1 be R_eal;
  assume e1 = 1;
  then for x being ExtReal holds x in S implies x <= e1 by A2,
XXREAL_1:1;
  then
A3: e1 is UpperBound of S by XXREAL_2:def 1;
  0. <= s by A2,A1,XXREAL_1:1;
  hence thesis by A3,A1,XXREAL_2:4,def 3;
end;
