theorem Th45:
  p <= r & s <= q implies ].r,s.[ c= [.p,q.[
proof
  assume that
A1: p <= r and
A2: s <= q;
  let t;
  assume
A3: t in ].r,s.[;
  then
A4: r < t by Th4;
A5: t < s by A3,Th4;
A6: p <= t by A1,A4,XXREAL_0:2;
  t < q by A2,A5,XXREAL_0:2;
  hence thesis by A6,Th3;
end;
