
theorem Th9:
  for a, b, c, d being Point of I[01] st a <= b & c <= d holds [:
  [.a,b.], [.c,d.]:] is compact non empty Subset of [:I[01], I[01]:]
proof
  let a, b, c, d be Point of I[01];
  [.a,b.] is Subset of I[01] & [.c,d.] is Subset of I[01] by BORSUK_4:18;
  then
A1: [:[.a,b.], [.c,d.]:] c= [:the carrier of I[01], the carrier of I[01]:]
  by ZFMISC_1:96;
  assume
A2: a <= b & c <= d;
  then a in [.a,b.] & c in [.c,d.] by XXREAL_1:1;
  then reconsider
  Ewa = [:[.a,b.], [.c,d.]:] as non empty Subset of [:I[01], I[01]
  :] by A1,BORSUK_1:def 2;
  [.a,b.] is compact Subset of I[01] & [.c,d.] is compact Subset of I[01]
  by A2,BORSUK_4:24;
  then Ewa is compact Subset of [:I[01], I[01]:] by BORSUK_3:23;
  hence thesis;
end;
