let r1, r2, x0, x1, x2, x3, x4, x5 be Real; :: thesis: for f1, f2 being Function of REAL,REAL holds [!((r1 (#) f1) + (r2 (#) f2)),x0,x1,x2,x3,x4,x5!] = (r1 * [!f1,x0,x1,x2,x3,x4,x5!]) + (r2 * [!f2,x0,x1,x2,x3,x4,x5!])
let f1, f2 be Function of REAL,REAL; :: thesis: [!((r1 (#) f1) + (r2 (#) f2)),x0,x1,x2,x3,x4,x5!] = (r1 * [!f1,x0,x1,x2,x3,x4,x5!]) + (r2 * [!f2,x0,x1,x2,x3,x4,x5!])
[!((r1 (#) f1) + (r2 (#) f2)),x0,x1,x2,x3,x4,x5!] = [!(r1 (#) f1),x0,x1,x2,x3,x4,x5!] + [!(r2 (#) f2),x0,x1,x2,x3,x4,x5!] by Th15
.= (r1 * [!f1,x0,x1,x2,x3,x4,x5!]) + [!(r2 (#) f2),x0,x1,x2,x3,x4,x5!] by Th14 ;
hence [!((r1 (#) f1) + (r2 (#) f2)),x0,x1,x2,x3,x4,x5!] = (r1 * [!f1,x0,x1,x2,x3,x4,x5!]) + (r2 * [!f2,x0,x1,x2,x3,x4,x5!]) by Th14; :: thesis: verum