
theorem Th14:
  for A, B being Subset of I[01], a, b, c being Real st a <
  b & b < c & A = [.a,b.[ & B = ].b,c.] holds A, B are_separated
proof
  let A, B be Subset of I[01], a, b, c be Real;
  assume that
A1: a < b and
A2: b < c and
A3: A = [.a,b.[ and
A4: B = ].b,c.];
  Cl A = [.a,b.] by A1,A3,Th12;
  hence Cl A misses B by A4,XXREAL_1:90;
  Cl B = [.b,c.] by A2,A4,Th11;
  hence thesis by A3,XXREAL_1:95;
end;
