let K be Field; :: thesis: for n, m, k being Element of NAT
for M1 being Matrix of n,m,K
for M2 being Matrix of m,k,K st width M1 = len M2 & 0 < len M1 & 0 < len M2 holds
M1 * M2 is Matrix of n,k,K
let n, m, k be Element of NAT ; :: thesis: for M1 being Matrix of n,m,K
for M2 being Matrix of m,k,K st width M1 = len M2 & 0 < len M1 & 0 < len M2 holds
M1 * M2 is Matrix of n,k,K
let M1 be Matrix of n,m,K; :: thesis: for M2 being Matrix of m,k,K st width M1 = len M2 & 0 < len M1 & 0 < len M2 holds
M1 * M2 is Matrix of n,k,K
let M2 be Matrix of m,k,K; :: thesis: ( width M1 = len M2 & 0 < len M1 & 0 < len M2 implies M1 * M2 is Matrix of n,k,K )
assume A1: ( width M1 = len M2 & 0 < len M1 & 0 < len M2 ) ; :: thesis: M1 * M2 is Matrix of n,k,K
then A2: ( len (M1 * M2) = len M1 & width (M1 * M2) = width M2 ) by MATRIX_3:def 4;
A3: len M1 = n by MATRIX_1:def 3;
width M1 = m by A1, MATRIX_1:def 3;
then width M2 = k by A1, MATRIX_1:20;
hence M1 * M2 is Matrix of n,k,K by A1, A2, A3, MATRIX_1:20; :: thesis: verum