let K be Field; :: thesis: 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; :: thesis: ( len M1 = len M2 & width M1 = width M2 & M2 - M1 = M2 implies M1 = 0. K,(len M1),(width M1) )
assume A1:
( len M1 = len M2 & width M1 = width M2 & M2 - M1 = M2 )
; :: thesis: M1 = 0. K,(len M1),(width M1)
per cases
( len M1 > 0 or len M1 = 0 )
by NAT_1:3;
suppose A2:
len M1 > 0
;
:: thesis: M1 = 0. K,(len M1),(width M1)then A3:
M2 is
Matrix of
len M1,
width M1,
K
by A1, MATRIX_1:20;
A4:
(
len (- M1) = len M1 &
width (- M1) = width M1 )
by MATRIX_3:def 2;
A5:
(
len (- M2) = len M2 &
width (- M2) = width M2 )
by MATRIX_3:def 2;
A6:
- M1 is
Matrix of
len M1,
width M1,
K
by A2, A4, MATRIX_1:20;
(M2 + (- M1)) + (- M2) = 0. K,
(len M1),
(width M1)
by A1, A3, MATRIX_3:7;
then
((- M1) + M2) + (- M2) = 0. K,
(len M1),
(width M1)
by A1, A4, MATRIX_3:4;
then
(- M1) + (M2 + (- M2)) = 0. K,
(len M1),
(width M1)
by A1, A4, A5, MATRIX_3:5;
then
(- M1) + (0. K,(len M1),(width M1)) = 0. K,
(len M1),
(width M1)
by A3, MATRIX_3:7;
then
- M1 = 0. K,
(len M1),
(width M1)
by A6, MATRIX_3:6;
then
M1 = - (0. K,(len M1),(width M1))
by Th1;
hence
M1 = 0. K,
(len M1),
(width M1)
by Th9;
:: thesis: verum end;