let x be Real; :: thesis: ( cos . x <> 0 implies ( sin / cos is_differentiable_in x & diff ((sin / cos),x) = 1 / ((cos . x) ^2) ) )
assume A1: cos . x <> 0 ; :: thesis: ( sin / cos is_differentiable_in x & diff ((sin / cos),x) = 1 / ((cos . x) ^2) )
A2: ( sin is_differentiable_in x & cos is_differentiable_in x ) by SIN_COS:63, SIN_COS:64;
then diff ((sin / cos),x) = (((diff (sin,x)) * (cos . x)) - ((diff (cos,x)) * (sin . x))) / ((cos . x) ^2) by A1, FDIFF_2:14
.= (((cos . x) * (cos . x)) - ((diff (cos,x)) * (sin . x))) / ((cos . x) ^2) by SIN_COS:64
.= (((cos . x) * (cos . x)) - ((- (sin . x)) * (sin . x))) / ((cos . x) ^2) by SIN_COS:63
.= (((cos . x) * (cos . x)) + ((sin . x) * (sin . x))) / ((cos . x) ^2)
.= 1 / ((cos . x) ^2) by SIN_COS:28 ;
hence ( sin / cos is_differentiable_in x & diff ((sin / cos),x) = 1 / ((cos . x) ^2) ) by A2, A1, FDIFF_2:14; :: thesis: verum