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 A1: ( width M1 = len M2 & len M1 > 0 & len M2 > 0 ) ; :: thesis: (0. K,(len M1),(width M1)) * M2 = 0. K,(len M1),(width M2)
A2: len (0. K,(len M1),(width M1)) = len M1 by MATRIX_1:def 3;
then A3: width (0. K,(len M1),(width M1)) = width M1 by A1, MATRIX_1:20;
then A4: len ((0. K,(len M1),(width M1)) * M2) = len (0. K,(len M1),(width M1)) by A1, MATRIX_3:def 4;
A5: width ((0. K,(len M1),(width M1)) * M2) = width M2 by A1, A3, MATRIX_3:def 4;
A6: len (- ((0. K,(len M1),(width M1)) * M2)) = len ((0. K,(len M1),(width M1)) * M2) by MATRIX_3:def 2;
A7: width (- ((0. K,(len M1),(width M1)) * M2)) = width ((0. K,(len M1),(width M1)) * M2) by MATRIX_3:def 2;
A8: (0. K,(len M1),(width M1)) * M2 = ((0. K,(len M1),(width M1)) + (0. K,(len M1),(width M1))) * M2 by MATRIX_3:6
.= ((0. K,(len M1),(width M1)) * M2) + ((0. K,(len M1),(width M1)) * M2) by A1, A2, A3, MATRIX_4:63 ;
set B = (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 A2, A4, A5, A8, 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 A6, A7, MATRIX_3:5
.= (0. K,(len M1),(width M1)) * M2 by A4, A7, MATRIX_4:20 ;
hence (0. K,(len M1),(width M1)) * M2 = 0. K,(len M1),(width M2) ; :: thesis: verum