let x0, x1 be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (cos x) ) & x0 <> x1 & cos x0 <> 0 & cos x1 <> 0 holds
[!f,x0,x1!] = ((2 * ((cos x1) - (cos x0))) / ((cos (x0 + x1)) + (cos (x0 - x1)))) / (x0 - x1)
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = 1 / (cos x) ) & x0 <> x1 & cos x0 <> 0 & cos x1 <> 0 implies [!f,x0,x1!] = ((2 * ((cos x1) - (cos x0))) / ((cos (x0 + x1)) + (cos (x0 - x1)))) / (x0 - x1) )
assume that
A1: for x being Real holds f . x = 1 / (cos x) and
x0 <> x1 and
A2: ( cos x0 <> 0 & cos x1 <> 0 ) ; :: thesis: [!f,x0,x1!] = ((2 * ((cos x1) - (cos x0))) / ((cos (x0 + x1)) + (cos (x0 - x1)))) / (x0 - x1)
( f . x0 = 1 / (cos x0) & f . x1 = 1 / (cos x1) ) by A1;
then [!f,x0,x1!] = (((1 * (cos x1)) - (1 * (cos x0))) / ((cos x0) * (cos x1))) / (x0 - x1) by A2, XCMPLX_1:130
.= (((cos x1) - (cos x0)) / ((1 / 2) * ((cos (x0 + x1)) + (cos (x0 - x1))))) / (x0 - x1) by SIN_COS4:32
.= ((((cos x1) - (cos x0)) / (1 / 2)) / ((cos (x0 + x1)) + (cos (x0 - x1)))) / (x0 - x1) by XCMPLX_1:78
.= (2 * (((cos x1) - (cos x0)) / ((cos (x0 + x1)) + (cos (x0 - x1))))) / (x0 - x1) ;
hence [!f,x0,x1!] = ((2 * ((cos x1) - (cos x0))) / ((cos (x0 + x1)) + (cos (x0 - x1)))) / (x0 - x1) ; :: thesis: verum