reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem
  for A being Subset of R^1, a, b, c being Real st a < b & b < c
  & A = RAT (a,b) \/ ]. b, c .[ \/ ]. c,+infty .[ holds Cl A = [. a,+infty .[
proof
  let A be Subset of R^1;
  let a, b, c be Real;
  assume that
A1: a < b and
A2: b < c;
  reconsider B = RAT (a,b) as Subset of R^1 by TOPMETR:17;
  reconsider C = ]. b, c .[ \/ ]. c,+infty .[ as Subset of R^1 by TOPMETR:17;
  assume A = RAT (a,b) \/ ]. b, c .[ \/ ]. c,+infty .[;
  then A = RAT (a,b) \/ C by XBOOLE_1:4;
  then Cl A = Cl B \/ Cl C by PRE_TOPC:20;
  then Cl A = Cl B \/ [. b,+infty .[ by A2,Th54;
  then Cl A = [. a, b .] \/ [. b,+infty .[ by A1,Th30;
  hence thesis by A1,Th10;
end;
