let a1, a2, a3, b1, b2, b3 be Real; :: thesis: for n being Nat
for x1, x2, x3 being Element of REAL n holds (((a1 * x1) + (a2 * x2)) + (a3 * x3)) + (((b1 * x1) + (b2 * x2)) + (b3 * x3)) = (((a1 + b1) * x1) + ((a2 + b2) * x2)) + ((a3 + b3) * x3)
let n be Nat; :: thesis: for x1, x2, x3 being Element of REAL n holds (((a1 * x1) + (a2 * x2)) + (a3 * x3)) + (((b1 * x1) + (b2 * x2)) + (b3 * x3)) = (((a1 + b1) * x1) + ((a2 + b2) * x2)) + ((a3 + b3) * x3)
let x1, x2, x3 be Element of REAL n; :: thesis: (((a1 * x1) + (a2 * x2)) + (a3 * x3)) + (((b1 * x1) + (b2 * x2)) + (b3 * x3)) = (((a1 + b1) * x1) + ((a2 + b2) * x2)) + ((a3 + b3) * x3)
thus (((a1 * x1) + (a2 * x2)) + (a3 * x3)) + (((b1 * x1) + (b2 * x2)) + (b3 * x3)) = (((a1 * x1) + (a2 * x2)) + ((b1 * x1) + (b2 * x2))) + ((a3 * x3) + (b3 * x3)) by Th16
.= (((a1 + b1) * x1) + ((a2 + b2) * x2)) + ((a3 * x3) + (b3 * x3)) by Th23
.= (((a1 + b1) * x1) + ((a2 + b2) * x2)) + ((a3 + b3) * x3) by EUCLID_4:7 ; :: thesis: verum