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

theorem Th29:
  for a, b, c being Real holds c in IRRAT (a,b) iff c is
  irrational & a < c & c < b
proof
  let a, b, c be Real;
  hereby
    assume
A1: c in IRRAT (a,b);
    then
A2: c in ]. a, b .[ by XBOOLE_0:def 4;
    c in IRRAT by A1,XBOOLE_0:def 4;
    hence c is irrational & a < c & c < b by A2,Th16,XXREAL_1:4;
  end;
  assume that
A3: c is irrational and
A4: a < c and
A5: c < b;
A6: c in ]. a, b .[ by A4,A5,XXREAL_1:4;
  c in IRRAT by A3,Th16;
  hence thesis by A6,XBOOLE_0:def 4;
end;
