let K be Field; :: thesis: for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 & M1 = M1 + M2 holds

M2 = 0. (K,(len M1),(width M1))

let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 & M1 = M1 + M2 implies M2 = 0. (K,(len M1),(width M1)) )

assume that

A1: len M1 = len M2 and

A2: width M1 = width M2 and

A3: M1 = M1 + M2 ; :: thesis: M2 = 0. (K,(len M1),(width M1))

0. (K,(len M1),(width M1)) = (M1 + M2) + (- M1) by A3, Th2;

then 0. (K,(len M1),(width M1)) = (M2 + M1) + (- M1) by A1, A2, MATRIX_3:2;

then 0. (K,(len M1),(width M1)) = M2 + (M1 + (- M1)) by A1, A2, MATRIX_3:3;

then A4: 0. (K,(len M1),(width M1)) = M2 + (0. (K,(len M1),(width M1))) by Th2;

M2 = 0. (K,(len M1),(width M1))

let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 & M1 = M1 + M2 implies M2 = 0. (K,(len M1),(width M1)) )

assume that

A1: len M1 = len M2 and

A2: width M1 = width M2 and

A3: M1 = M1 + M2 ; :: thesis: M2 = 0. (K,(len M1),(width M1))

0. (K,(len M1),(width M1)) = (M1 + M2) + (- M1) by A3, Th2;

then 0. (K,(len M1),(width M1)) = (M2 + M1) + (- M1) by A1, A2, MATRIX_3:2;

then 0. (K,(len M1),(width M1)) = M2 + (M1 + (- M1)) by A1, A2, MATRIX_3:3;

then A4: 0. (K,(len M1),(width M1)) = M2 + (0. (K,(len M1),(width M1))) by Th2;