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

theorem
  (]. -infty, 1 .[ \/ ]. 1,+infty .[) /\ (]. -infty, 2 .] \/ IRRAT(2,4)
  \/ {4} \/ {5}) = ]. -infty, 1 .[ \/ ]. 1, 2 .] \/ IRRAT(2,4) \/ {4} \/ {5}
proof
  (]. -infty, 1 .[ \/ ]. 1,+infty .[) /\ (]. -infty, 2 .] \/ IRRAT(2,4) \/
{4} \/ {5}) = ]. -infty, 1 .[ \/ (]. 1,+infty .[ /\ (]. -infty, 2 .] \/ IRRAT(2
  ,4) \/ {4} \/ {5})) by Lm9,XBOOLE_1:23
    .= ]. -infty, 1 .[ \/ (]. 1, 2 .] \/ IRRAT(2,4) \/ ({4} \/ {5})) by Lm10,
XBOOLE_1:4
    .= ]. -infty, 1 .[ \/ ]. 1, 2 .] \/ IRRAT(2,4) \/ ({4} \/ {5}) by
XBOOLE_1:113;
  hence thesis by XBOOLE_1:4;
end;
