let x0, x1, x2, x3, x4 be Real; :: thesis: for f1, f2 being Function of REAL,REAL holds [!(f1 + f2),x0,x1,x2,x3,x4!] = [!f1,x0,x1,x2,x3,x4!] + [!f2,x0,x1,x2,x3,x4!]
let f1, f2 be Function of REAL,REAL; :: thesis: [!(f1 + f2),x0,x1,x2,x3,x4!] = [!f1,x0,x1,x2,x3,x4!] + [!f2,x0,x1,x2,x3,x4!]
[!(f1 + f2),x0,x1,x2,x3,x4!] = (([!f1,x0,x1,x2,x3!] + [!f2,x0,x1,x2,x3!]) - [!(f1 + f2),x1,x2,x3,x4!]) / (x0 - x4) by Th9
.= (([!f1,x0,x1,x2,x3!] + [!f2,x0,x1,x2,x3!]) - ([!f1,x1,x2,x3,x4!] + [!f2,x1,x2,x3,x4!])) / (x0 - x4) by Th9
.= (([!f1,x0,x1,x2,x3!] - [!f1,x1,x2,x3,x4!]) + ([!f2,x0,x1,x2,x3!] - [!f2,x1,x2,x3,x4!])) / (x0 - x4) ;
hence [!(f1 + f2),x0,x1,x2,x3,x4!] = [!f1,x0,x1,x2,x3,x4!] + [!f2,x0,x1,x2,x3,x4!] by XCMPLX_1:62; :: thesis: verum