
theorem Th6:
  for a, b being Real st a in ].0,PI/2.[ & b in ].0,PI/2.[
  holds a < b iff sin a < sin b
proof
  let a, b be Real;
  assume a in ].0,PI/2.[ & b in ].0,PI/2.[;
  then
A1: a in ].0,PI/2.[ /\ dom sin & b in ].0,PI/2.[ /\ dom sin by SIN_COS:24
,XBOOLE_0:def 4;
A2: sin a = sin.a & sin b = sin.b by SIN_COS:def 17;
  hence a < b implies sin a < sin b by A1,RFUNCT_2:20,SIN_COS2:2;
  assume
A3: sin a < sin b;
  assume a >= b;
  then a > b by A3,XXREAL_0:1;
  hence contradiction by A2,A1,A3,RFUNCT_2:20,SIN_COS2:2;
end;
