for x being Real st x in ].(PI / 2),PI .[ holds
diff sec ,x > 0

proof

let x be Real; :: thesis: ( x in ].(PI / 2),PI .[ implies diff sec ,x > 0 )
assume A1: x in ].(PI / 2),PI .[ ; :: thesis: diff sec ,x > 0
PI <= (3 / 2) * PI by XREAL_1:153;
then ].(PI / 2),PI .[ c= ].(PI / 2),((3 / 2) * PI ).[ by XXREAL_1:46;
then A2: cos . x < 0 by A1, COMPTRIG:29;
].(PI / 2),PI .[ c= ].0 ,PI .[ by XXREAL_1:46;
then sin . x > 0 by A1, COMPTRIG:23;
then (sin . x) / ((cos . x) ^2 ) > 0 / ((cos . x) ^2 ) by A2;
hence diff sec ,x > 0 by A1, Th6; :: thesis: verum

end;

hence sec | ].(PI / 2),PI .[ is increasing by Lm11, Th2, Th6, ROLLE:9, XBOOLE_1:1; :: thesis: verum