A1: for x being Real st x in ].(PI / 2),((3 / 2) * PI ).[ holds
diff sin ,x < 0

proof

let x be Real; :: thesis: ( x in ].(PI / 2),((3 / 2) * PI ).[ implies diff sin ,x < 0 )
assume x in ].(PI / 2),((3 / 2) * PI ).[ ; :: thesis: diff sin ,x < 0
then 0 > cos . x by Th29;
hence diff sin ,x < 0 by SIN_COS:73; :: thesis: verum

end;

].(PI / 2),((3 / 2) * PI ).[ is open by RCOMP_1:25;
hence sin | ].(PI / 2),((3 / 2) * PI ).[ is decreasing by A1, FDIFF_1:34, ROLLE:10, SIN_COS:27, SIN_COS:73; :: thesis: verum