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

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