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

theorem Th28:
  for a, b, c being Real holds c in RAT (a,b) iff c is
  rational & a < c & c < b
proof
  let a, b, c be Real;
  hereby
    assume
A1: c in RAT (a,b);
    then c in ]. a, b .[ by XBOOLE_0:def 4;
    hence c is rational & a < c & c < b by A1,XXREAL_1:4;
  end;
  assume that
A2: c is rational and
A3: a < c and
A4: c < b;
A5: c in RAT by A2,RAT_1:def 2;
  c in ]. a, b .[ by A3,A4,XXREAL_1:4;
  hence thesis by A5,XBOOLE_0:def 4;
end;
