let h, x0, x1 be Real; :: thesis: for f, g being Function of REAL,REAL st ( for x being Real holds f . x = (cD (g,h)) . x ) holds
[!f,x0,x1!] = [!g,(x0 + (h / 2)),(x1 + (h / 2))!] - [!g,(x0 - (h / 2)),(x1 - (h / 2))!]
let f, g be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = (cD (g,h)) . x ) implies [!f,x0,x1!] = [!g,(x0 + (h / 2)),(x1 + (h / 2))!] - [!g,(x0 - (h / 2)),(x1 - (h / 2))!] )
assume A1: for x being Real holds f . x = (cD (g,h)) . x ; :: thesis: [!f,x0,x1!] = [!g,(x0 + (h / 2)),(x1 + (h / 2))!] - [!g,(x0 - (h / 2)),(x1 - (h / 2))!]
[!f,x0,x1!] = (((cD (g,h)) . x0) - (f . x1)) / (x0 - x1) by A1
.= (((cD (g,h)) . x0) - ((cD (g,h)) . x1)) / (x0 - x1) by A1
.= (((g . (x0 + (h / 2))) - (g . (x0 - (h / 2)))) - ((cD (g,h)) . x1)) / (x0 - x1) by DIFF_1:5
.= (((g . (x0 + (h / 2))) - (g . (x0 - (h / 2)))) - ((g . (x1 + (h / 2))) - (g . (x1 - (h / 2))))) / (x0 - x1) by DIFF_1:5
.= [!g,(x0 + (h / 2)),(x1 + (h / 2))!] - [!g,(x0 - (h / 2)),(x1 - (h / 2))!] ;
hence [!f,x0,x1!] = [!g,(x0 + (h / 2)),(x1 + (h / 2))!] - [!g,(x0 - (h / 2)),(x1 - (h / 2))!] ; :: thesis: verum