let n be Element of NAT ; :: thesis: for r, x being Real st 0 < r holds
(Maclaurin exp_R ,].(- r),r.[,x) . n = (x |^ n) / (n ! )
let r, x be Real; :: thesis: ( 0 < r implies (Maclaurin exp_R ,].(- r),r.[,x) . n = (x |^ n) / (n ! ) )
assume A1: r > 0 ; :: thesis: (Maclaurin exp_R ,].(- r),r.[,x) . n = (x |^ n) / (n ! )
abs (0 - 0 ) = 0 by ABSVALUE:7;
then 0 in ].(0 - r),(0 + r).[ by A1, RCOMP_1:8;
then A2: 0 in dom (exp_R | ].(- r),r.[) by Th5;
(Maclaurin exp_R ,].(- r),r.[,x) . n = ((((diff exp_R ,].(- r),r.[) . n) . 0 ) * ((x - 0 ) |^ n)) / (n ! ) by TAYLOR_1:def 7
.= (((exp_R | ].(- r),r.[) . 0 ) * (x |^ n)) / (n ! ) by Th6
.= ((exp_R . 0 ) * (x |^ n)) / (n ! ) by A2, FUNCT_1:70
.= (x |^ n) / (n ! ) by SIN_COS2:13 ;
hence (Maclaurin exp_R ,].(- r),r.[,x) . n = (x |^ n) / (n ! ) ; :: thesis: verum