let p, q, r be Element of (Formal-Series L); :: according to VECTSP_1:def 3 :: thesis: (q + r) * p = (q * p) + (r * p)

reconsider p1 = p, q1 = q, r1 = r as sequence of L by Def2;

A2: ( q * p = q1 *' p1 & r * p = r1 *' p1 ) by Def2;

q + r = q1 + r1 by Def2;

hence (q + r) * p = (q1 + r1) *' p1 by Def2

.= (q1 *' p1) + (r1 *' p1) by POLYNOM3:32

.= (q * p) + (r * p) by A2, Def2 ;

:: thesis: verum

reconsider p1 = p, q1 = q, r1 = r as sequence of L by Def2;

A2: ( q * p = q1 *' p1 & r * p = r1 *' p1 ) by Def2;

q + r = q1 + r1 by Def2;

hence (q + r) * p = (q1 + r1) *' p1 by Def2

.= (q1 *' p1) + (r1 *' p1) by POLYNOM3:32

.= (q * p) + (r * p) by A2, Def2 ;

:: thesis: verum