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 that

A1: ( len M1 = len M2 & width M1 = width M2 ) and

A2: M2 - M1 = M2 ; :: thesis: M1 = 0. (K,(len M1),(width M1))

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 that

A1: ( len M1 = len M2 & width M1 = width M2 ) and

A2: M2 - M1 = M2 ; :: thesis: M1 = 0. (K,(len M1),(width M1))

per cases
( len M1 > 0 or len M1 = 0 )
by NAT_1:3;

end;

suppose A3:
len M1 > 0
; :: thesis: 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; :: thesis: verum

end;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; :: thesis: verum