let K be Field; :: thesis: 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; :: thesis: ( 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 ; :: thesis: (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)) ; :: thesis: verum