now

let r1, r2 be Real; :: thesis: ( r1 in [.(PI / 2),((3 / 2) * PI ).] /\ (dom sin ) & r2 in [.(PI / 2),((3 / 2) * PI ).] /\ (dom sin ) & r1 < r2 implies sin . r2 < sin . r1 )
assume that
A1: r1 in [.(PI / 2),((3 / 2) * PI ).] /\ (dom sin ) and
A2: r2 in [.(PI / 2),((3 / 2) * PI ).] /\ (dom sin ) and
A3: r1 < r2 ; :: thesis: sin . r2 < sin . r1
A4: ( r1 in [.(PI / 2),((3 / 2) * PI ).] & r1 in dom sin & r2 in [.(PI / 2),((3 / 2) * PI ).] & r2 in dom sin ) by A1, A2, XBOOLE_0:def 4;
then A5: ( PI / 2 <= r1 & r1 <= (3 / 2) * PI & PI / 2 <= r2 & r2 <= (3 / 2) * PI ) by XXREAL_1:1;
abs (sin r1) <= 1 by SIN_COS:30;
then abs (sin . r1) <= 1 by SIN_COS:def 21;
then A6: sin . r1 >= - 1 by ABSVALUE:12;
set r3 = (r1 + r2) / 2;
abs (sin ((r1 + r2) / 2)) <= 1 by SIN_COS:30;
then abs (sin . ((r1 + r2) / 2)) <= 1 by SIN_COS:def 21;
then A7: ( sin . ((r1 + r2) / 2) >= - 1 & sin . ((r1 + r2) / 2) <= 1 ) by ABSVALUE:12;
abs (sin r2) <= 1 by SIN_COS:30;
then abs (sin . r2) <= 1 by SIN_COS:def 21;
then A8: sin . r2 <= 1 by ABSVALUE:12;
A9: ( r1 < (r1 + r2) / 2 & (r1 + r2) / 2 < r2 ) by A3, XREAL_1:228;
then ( PI / 2 < (r1 + r2) / 2 & (r1 + r2) / 2 < (3 / 2) * PI ) by A5, XXREAL_0:2;
then ( (r1 + r2) / 2 in ].(PI / 2),((3 / 2) * PI ).[ & (r1 + r2) / 2 in dom sin ) by SIN_COS:27, XXREAL_1:4;
then A10: (r1 + r2) / 2 in ].(PI / 2),((3 / 2) * PI ).[ /\ (dom sin ) by XBOOLE_0:def 4;

now

per cases ( PI / 2 < r1 or PI / 2 = r1 ) by A5, XXREAL_0:1;

suppose A11: PI / 2 < r1 ; :: thesis: sin . r2 < sin . r1

then A12: PI / 2 < r2 by A3, XXREAL_0:2;

now

per cases ( r2 < (3 / 2) * PI or r2 = (3 / 2) * PI ) by A5, XXREAL_0:1;

suppose A13: r2 < (3 / 2) * PI ; :: thesis: sin . r2 < sin . r1

then r1 < (3 / 2) * PI by A3, XXREAL_0:2;
then ( r1 in ].(PI / 2),((3 / 2) * PI ).[ & r2 in ].(PI / 2),((3 / 2) * PI ).[ ) by A11, A12, A13, XXREAL_1:4;
then ( r1 in ].(PI / 2),((3 / 2) * PI ).[ /\ (dom sin ) & r2 in ].(PI / 2),((3 / 2) * PI ).[ /\ (dom sin ) ) by A4, XBOOLE_0:def 4;
hence sin . r2 < sin . r1 by A3, Th36, RFUNCT_2:44; :: thesis: verum

end;

suppose A14: r2 = (3 / 2) * PI ; :: thesis: not sin . r2 >= sin . r1

then r1 in ].(PI / 2),((3 / 2) * PI ).[ by A3, A11, XXREAL_1:4;
then r1 in ].(PI / 2),((3 / 2) * PI ).[ /\ (dom sin ) by A4, XBOOLE_0:def 4;
then A15: sin . ((r1 + r2) / 2) < sin . r1 by A9, A10, Th36, RFUNCT_2:44;
assume sin . r2 >= sin . r1 ; :: thesis: contradiction
hence contradiction by A6, A7, A14, A15, SIN_COS:81, XXREAL_0:1; :: thesis: verum

end;

end;

end;

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

end;

suppose A16: PI / 2 = r1 ; :: thesis: sin . r2 < sin . r1

now

per cases ( r2 < (3 / 2) * PI or r2 = (3 / 2) * PI ) by A5, XXREAL_0:1;

suppose r2 < (3 / 2) * PI ; :: thesis: not sin . r2 >= sin . r1

then r2 in ].(PI / 2),((3 / 2) * PI ).[ by A3, A16, XXREAL_1:4;
then r2 in ].(PI / 2),((3 / 2) * PI ).[ /\ (dom sin ) by A4, XBOOLE_0:def 4;
then A17: sin . r2 < sin . ((r1 + r2) / 2) by A9, A10, Th36, RFUNCT_2:44;
assume sin . r2 >= sin . r1 ; :: thesis: contradiction
hence contradiction by A7, A8, A16, A17, SIN_COS:81, XXREAL_0:1; :: thesis: verum

end;

suppose r2 = (3 / 2) * PI ; :: thesis: sin . r2 < sin . r1

hence sin . r2 < sin . r1 by A16, SIN_COS:81; :: thesis: verum

end;

end;

end;

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

end;

end;

end;

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

end;

hence sin | [.(PI / 2),((3 / 2) * PI ).] is decreasing by RFUNCT_2:44; :: thesis: verum