reserve T for TopSpace,
  A, B for Subset of T;

theorem
  for a,b being Real st a < b holds REAL = ]. -infty,a.[ \/ [.a,b
  .] \/ ].b,+infty .[
proof
  let a,b be Real;
  assume
A1: a < b;
  REAL = (REAL \ {a}) \/ {a} by XBOOLE_1:45
    .= (]. -infty,a.[ \/ ].a,+infty .[) \/ {a} by XXREAL_1:389
    .= ]. -infty,a.[ \/ (].a,+infty .[ \/ {a}) by XBOOLE_1:4
    .= ]. -infty,a.[ \/ [.a,+infty .[ by BORSUK_5:43
    .= ]. -infty,a.[ \/ ([.a,b.] \/ [.b,+infty .[) by A1,BORSUK_5:11
    .= ]. -infty,a.[ \/ ([.a,b.] \/ ({b} \/ ].b,+infty .[)) by BORSUK_5:43
    .= ]. -infty,a.[ \/ ([.a,b.] \/ {b} \/ ].b,+infty .[) by XBOOLE_1:4
    .= ]. -infty,a.[ \/ ([.a,b.] \/ {b}) \/ ].b,+infty .[ by XBOOLE_1:4;
  hence thesis by A1,Lm1;
end;
