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