let r1, r2, x0, x1, x2 be Real; :: thesis: for f1, f2 being Function of REAL,REAL holds [!((r1 (#) f1) + (r2 (#) f2)),x0,x1,x2!] = (r1 * [!f1,x0,x1,x2!]) + (r2 * [!f2,x0,x1,x2!])
let f1, f2 be Function of REAL,REAL; :: thesis: [!((r1 (#) f1) + (r2 (#) f2)),x0,x1,x2!] = (r1 * [!f1,x0,x1,x2!]) + (r2 * [!f2,x0,x1,x2!])
[!((r1 (#) f1) + (r2 (#) f2)),x0,x1,x2!] = [!(r1 (#) f1),x0,x1,x2!] + [!(r2 (#) f2),x0,x1,x2!] by Th6
.= (r1 * [!f1,x0,x1,x2!]) + [!(r2 (#) f2),x0,x1,x2!] by Th5 ;
hence [!((r1 (#) f1) + (r2 (#) f2)),x0,x1,x2!] = (r1 * [!f1,x0,x1,x2!]) + (r2 * [!f2,x0,x1,x2!]) by Th5; :: thesis: verum