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

theorem Th58:
  for a, b being Real st a < b holds [.a,+infty .[ \ (]. a,
  b .[ \/ ]. b,+infty .[) = {a} \/ {b}
proof
  let a, b be Real;
A1: not b in ]. a,b .[ \/ ]. b,+infty .[ by XXREAL_1:205;
  assume
A2: a < b;
  then
A3: not a in ]. a,b .[ \/ ]. b,+infty .[ by XXREAL_1:223;
  [. a,+infty .[ = [. a,b .] \/ [. b,+infty .[ by A2,Th10
    .= {a, b} \/ ]. a,b .[ \/ [. b,+infty .[ by A2,XXREAL_1:128
    .= {a, b} \/ ]. a,b .[ \/ ({b} \/ ]. b,+infty .[) by Th42
    .= {a, b} \/ ]. a,b .[ \/ {b} \/ ]. b,+infty .[ by XBOOLE_1:4
    .= {a, b} \/ {b} \/ ]. a,b .[ \/ ]. b,+infty .[ by XBOOLE_1:4
    .= {a, b} \/ ]. a,b .[ \/ ]. b,+infty .[ by ZFMISC_1:9
    .= {a, b} \/ (]. a,b .[ \/ ]. b,+infty .[) by XBOOLE_1:4;
  then [.a,+infty .[ \ (]. a, b .[ \/ ]. b,+infty .[) = {a, b} by A3,A1,
XBOOLE_1:88,ZFMISC_1:51;
  hence thesis by ENUMSET1:1;
end;
