now

let r1, r2 be Real; :: thesis: ( r1 in [.0 ,PI .] /\ (dom cos ) & r2 in [.0 ,PI .] /\ (dom cos ) & r1 < r2 implies cos . r2 < cos . r1 )
assume that
A1: r1 in [.0 ,PI .] /\ (dom cos ) and
A2: r2 in [.0 ,PI .] /\ (dom cos ) and
A3: r1 < r2 ; :: thesis: cos . r2 < cos . r1
A4: r1 in dom cos by A1, XBOOLE_0:def 4;
abs (cos r2) <= 1 by SIN_COS:30;
then abs (cos . r2) <= 1 by SIN_COS:def 23;
then A5: cos . r2 <= 1 by ABSVALUE:12;
abs (cos r1) <= 1 by SIN_COS:30;
then abs (cos . r1) <= 1 by SIN_COS:def 23;
then A6: cos . r1 >= - 1 by ABSVALUE:12;
r2 in [.0 ,PI .] by A2, XBOOLE_0:def 4;
then A7: r2 <= PI by XXREAL_1:1;
set r3 = (r1 + r2) / 2;
A8: r1 < (r1 + r2) / 2 by A3, XREAL_1:228;
abs (cos ((r1 + r2) / 2)) <= 1 by SIN_COS:30;
then A9: abs (cos . ((r1 + r2) / 2)) <= 1 by SIN_COS:def 23;
then A10: cos . ((r1 + r2) / 2) <= 1 by ABSVALUE:12;
A11: r2 in dom cos by A2, XBOOLE_0:def 4;
A12: r1 in [.0 ,PI .] by A1, XBOOLE_0:def 4;
then A13: 0 < (r1 + r2) / 2 by A8, XXREAL_1:1;
A14: (r1 + r2) / 2 < r2 by A3, XREAL_1:228;
then (r1 + r2) / 2 < PI by A7, XXREAL_0:2;
then (r1 + r2) / 2 in ].0 ,PI .[ by A13, XXREAL_1:4;
then A15: (r1 + r2) / 2 in ].0 ,PI .[ /\ (dom cos ) by SIN_COS:27, XBOOLE_0:def 4;
A16: cos . ((r1 + r2) / 2) >= - 1 by A9, ABSVALUE:12;

now

per cases ( 0 < r1 or 0 = r1 ) by A12, XXREAL_1:1;

suppose A17: 0 < r1 ; :: thesis: cos . r2 < cos . r1

now

per cases ( r2 < PI or r2 = PI ) by A7, XXREAL_0:1;

suppose A18: r2 < PI ; :: thesis: cos . r2 < cos . r1

then r1 < PI by A3, XXREAL_0:2;
then r1 in ].0 ,PI .[ by A17, XXREAL_1:4;
then A19: r1 in ].0 ,PI .[ /\ (dom cos ) by A4, XBOOLE_0:def 4;
r2 in ].0 ,PI .[ by A3, A17, A18, XXREAL_1:4;
then r2 in ].0 ,PI .[ /\ (dom cos ) by A11, XBOOLE_0:def 4;
hence cos . r2 < cos . r1 by A3, A19, Th37, RFUNCT_2:44; :: thesis: verum

end;

suppose A20: r2 = PI ; :: thesis: not cos . r2 >= cos . r1

then r1 in ].0 ,PI .[ by A3, A17, XXREAL_1:4;
then r1 in ].0 ,PI .[ /\ (dom cos ) by A4, XBOOLE_0:def 4;
then A21: cos . ((r1 + r2) / 2) < cos . r1 by A8, A15, Th37, RFUNCT_2:44;
assume cos . r2 >= cos . r1 ; :: thesis: contradiction
hence contradiction by A6, A16, A20, A21, SIN_COS:81, XXREAL_0:1; :: thesis: verum

end;

end;

end;

hence cos . r2 < cos . r1 ; :: thesis: verum

end;

suppose A22: 0 = r1 ; :: thesis: cos . r2 < cos . r1

now

per cases ( r2 < PI or r2 = PI ) by A7, XXREAL_0:1;

suppose r2 < PI ; :: thesis: not cos . r2 >= cos . r1

then r2 in ].0 ,PI .[ by A3, A22, XXREAL_1:4;
then r2 in ].0 ,PI .[ /\ (dom cos ) by A11, XBOOLE_0:def 4;
then A23: cos . r2 < cos . ((r1 + r2) / 2) by A14, A15, Th37, RFUNCT_2:44;
assume cos . r2 >= cos . r1 ; :: thesis: contradiction
hence contradiction by A10, A5, A22, A23, SIN_COS:33, XXREAL_0:1; :: thesis: verum

end;

suppose r2 = PI ; :: thesis: cos . r2 < cos . r1

hence cos . r2 < cos . r1 by A22, SIN_COS:33, SIN_COS:81; :: thesis: verum

end;

end;

end;

hence cos . r2 < cos . r1 ; :: thesis: verum

end;

end;

end;

hence cos . r2 < cos . r1 ; :: thesis: verum

end;

hence cos | [.0 ,PI .] is decreasing by RFUNCT_2:44; :: thesis: verum