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

theorem Th65:
  for A being Subset of R^1, a, b, c being Real st a < b &
b < c & A = ]. -infty, a .[ \/ ]. a, b .] \/ IRRAT(b,c) \/ {c} holds Cl A = ].
  -infty, c .]
proof
  let A be Subset of R^1, a, b, c be Real;
  assume that
A1: a < b and
A2: b < c and
A3: A = ]. -infty, a .[ \/ ]. a, b .] \/ IRRAT(b,c) \/ {c};
  reconsider B = ]. -infty, a .[, C = ]. a, b .], D = IRRAT(b,c), E = {c} as
  Subset of R^1 by TOPMETR:17;
A4: c in ]. -infty, c .] by XXREAL_1:234;
  Cl A = Cl (B \/ C \/ D) \/ Cl E by A3,PRE_TOPC:20
    .= Cl (B \/ C \/ D) \/ E by Th37
    .= Cl (B \/ C) \/ Cl D \/ E by PRE_TOPC:20
    .= ]. -infty, b .] \/ Cl D \/ E by A1,Th64
    .= ]. -infty, b .] \/ [. b,c .] \/ E by A2,Th31
    .= ]. -infty, c .] \/ E by A2,Th11
    .= ]. -infty, c .] by A4,ZFMISC_1:40;
  hence thesis;
end;
