A1: sin is_differentiable_on ].0 ,1.[ by Th73, FDIFF_1:34;
A2: cos is_differentiable_on ].0 ,1.[ by Th72, FDIFF_1:34;
A3: for th being Real st th in ].0 ,1.[ holds
cos . th <> 0

proof

let th be Real; :: thesis: ( th in ].0 ,1.[ implies cos . th <> 0 )
assume A4: th in ].0 ,1.[ ; :: thesis: cos . th <> 0
].0 ,1.[ c= [.0 ,1.] by XXREAL_1:25;
hence cos . th <> 0 by A4, Th74; :: thesis: verum

end;

for th being real number st th in ].0 ,1.[ holds
diff tan ,th > 0

proof

let th be real number ; :: thesis: ( th in ].0 ,1.[ implies diff tan ,th > 0 )
assume A5: th in ].0 ,1.[ ; :: thesis: diff tan ,th > 0
A6: th is Real by XREAL_0:def 1;
A7: cos is_differentiable_in th by Th68;
A8: sin is_differentiable_in th by Th69;
A9: cos . th <> 0 by A3, A5;
then A10: diff tan ,th = (((diff sin ,th) * (cos . th)) - ((diff cos ,th) * (sin . th))) / ((cos . th) ^2 ) by A6, A7, A8, FDIFF_2:14
.= (((cos . th) * (cos . th)) - ((diff cos ,th) * (sin . th))) / ((cos . th) ^2 ) by Th69
.= (((cos . th) * (cos . th)) - ((- (sin . th)) * (sin . th))) / ((cos . th) ^2 ) by Th68
.= (((cos . th) ^2 ) + ((sin . th) * (sin . th))) / ((cos . th) ^2 )
.= 1 / ((cos . th) ^2 ) by Th31 ;
(cos . th) ^2 > 0 by A9, SQUARE_1:74;
hence diff tan ,th > 0 by A10; :: thesis: verum

end;

hence ( tan is_differentiable_on ].0 ,1.[ & ( for th being real number st th in ].0 ,1.[ holds
diff tan ,th > 0 ) ) by A1, A2, A3, FDIFF_2:21; :: thesis: verum