let K be Field; for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 & M2 - M1 = M2 holds
M1 = 0. (K,(len M1),(width M1))
let M1, M2 be Matrix of K; ( len M1 = len M2 & width M1 = width M2 & M2 - M1 = M2 implies M1 = 0. (K,(len M1),(width M1)) )
assume that
A1:
( len M1 = len M2 & width M1 = width M2 )
and
A2:
M2 - M1 = M2
; M1 = 0. (K,(len M1),(width M1))
per cases
( len M1 > 0 or len M1 = 0 )
by NAT_1:3;
suppose A3:
len M1 > 0
;
M1 = 0. (K,(len M1),(width M1))A4:
(
len (- M1) = len M1 &
width (- M1) = width M1 )
by MATRIX_3:def 2;
A5:
M2 is
Matrix of
len M1,
width M1,
K
by A1, A3, MATRIX_0:20;
then
(M2 + (- M1)) + (- M2) = 0. (
K,
(len M1),
(width M1))
by A2, MATRIX_3:5;
then
((- M1) + M2) + (- M2) = 0. (
K,
(len M1),
(width M1))
by A1, A4, MATRIX_3:2;
then
(- M1) + (M2 + (- M2)) = 0. (
K,
(len M1),
(width M1))
by A1, A4, MATRIX_3:3;
then A6:
(- M1) + (0. (K,(len M1),(width M1))) = 0. (
K,
(len M1),
(width M1))
by A5, MATRIX_3:5;
- M1 is
Matrix of
len M1,
width M1,
K
by A3, A4, MATRIX_0:20;
then
- M1 = 0. (
K,
(len M1),
(width M1))
by A6, MATRIX_3:4;
then
M1 = - (0. (K,(len M1),(width M1)))
by Th1;
hence
M1 = 0. (
K,
(len M1),
(width M1))
by Th9;
verum end; end;