let K be Field; for M1, M2 being Matrix of K st width M1 = len M2 & len M1 > 0 & len M2 > 0 holds
(0. (K,(len M1),(width M1))) * M2 = 0. (K,(len M1),(width M2))
let M1, M2 be Matrix of K; ( width M1 = len M2 & len M1 > 0 & len M2 > 0 implies (0. (K,(len M1),(width M1))) * M2 = 0. (K,(len M1),(width M2)) )
assume that
A1:
width M1 = len M2
and
A2:
len M1 > 0
and
A3:
len M2 > 0
; (0. (K,(len M1),(width M1))) * M2 = 0. (K,(len M1),(width M2))
A4:
len (0. (K,(len M1),(width M1))) = len M1
by MATRIX_0:def 2;
then A5:
width (0. (K,(len M1),(width M1))) = width M1
by A2, MATRIX_0:20;
then A6:
len ((0. (K,(len M1),(width M1))) * M2) = len (0. (K,(len M1),(width M1)))
by A1, MATRIX_3:def 4;
A7:
width ((0. (K,(len M1),(width M1))) * M2) = width M2
by A1, A5, MATRIX_3:def 4;
set B = (0. (K,(len M1),(width M1))) * M2;
A8:
width (- ((0. (K,(len M1),(width M1))) * M2)) = width ((0. (K,(len M1),(width M1))) * M2)
by MATRIX_3:def 2;
(0. (K,(len M1),(width M1))) * M2 =
((0. (K,(len M1),(width M1))) + (0. (K,(len M1),(width M1)))) * M2
by MATRIX_3:4
.=
((0. (K,(len M1),(width M1))) * M2) + ((0. (K,(len M1),(width M1))) * M2)
by A1, A2, A3, A4, A5, MATRIX_4:63
;
then
( len (- ((0. (K,(len M1),(width M1))) * M2)) = len ((0. (K,(len M1),(width M1))) * M2) & 0. (K,(len M1),(width M2)) = (((0. (K,(len M1),(width M1))) * M2) + ((0. (K,(len M1),(width M1))) * M2)) + (- ((0. (K,(len M1),(width M1))) * M2)) )
by A4, A6, A7, MATRIX_3:def 2, MATRIX_4:2;
then 0. (K,(len M1),(width M2)) =
((0. (K,(len M1),(width M1))) * M2) + (((0. (K,(len M1),(width M1))) * M2) - ((0. (K,(len M1),(width M1))) * M2))
by A8, MATRIX_3:3
.=
(0. (K,(len M1),(width M1))) * M2
by A6, A8, MATRIX_4:20
;
hence
(0. (K,(len M1),(width M1))) * M2 = 0. (K,(len M1),(width M2))
; verum