let K be Field; for n, m, k being 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 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 M1 be 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 M2 be Matrix of m,k,K; ( width M1 = len M2 & 0 < len M1 & 0 < len M2 implies M1 * M2 is Matrix of n,k,K )
assume that
A1:
width M1 = len M2
and
A2:
0 < len M1
and
A3:
0 < len M2
; M1 * M2 is Matrix of n,k,K
width M1 = m
by A1, MATRIX_0:def 2;
then A4:
( len M1 = n & width M2 = k )
by A1, A3, MATRIX_0:20, MATRIX_0:def 2;
( len (M1 * M2) = len M1 & width (M1 * M2) = width M2 )
by A1, MATRIX_3:def 4;
hence
M1 * M2 is Matrix of n,k,K
by A2, A4, MATRIX_0:20; verum