let K be Ring; :: thesis: for M1, M2, M3 being Matrix of K st len M1 = len M2 & len M2 = len M3 & width M1 = width M2 & width M2 = width M3 & M3 - M1 = M3 - M2 holds
M1 = M2
let M1, M2, M3 be Matrix of K; :: thesis: ( len M1 = len M2 & len M2 = len M3 & width M1 = width M2 & width M2 = width M3 & M3 - M1 = M3 - M2 implies M1 = M2 )
assume that
A1: len M1 = len M2 and
A2: len M2 = len M3 and
A3: width M1 = width M2 and
A4: width M2 = width M3 and
A5: M3 - M1 = M3 - M2 ; :: thesis: M1 = M2

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

suppose A6: len M1 > 0 ; :: thesis: M1 = M2

then A7: M3 is Matrix of len M1, width M1,K by A1, A2, A3, A4, MATRIX_0:20;
A8: ( len (- M2) = len M2 & width (- M2) = width M2 ) by MATRIX_3:def 2;
then A9: - M2 is Matrix of len M1, width M1,K by A1, A3, A6, MATRIX_0:20;
A10: ( len (- M1) = len M1 & width (- M1) = width M1 ) by MATRIX_3:def 2;
then (- M1) + M3 = M3 + (- M2) by A1, A2, A3, A4, A5, MATRIX_3:2;
then (- M1) + M3 = (- M2) + M3 by A2, A4, A8, MATRIX_3:2;
then ((- M1) + M3) + (- M3) = (- M2) + (M3 + (- M3)) by A2, A4, A8, MATRIX_3:3;
then ((- M1) + M3) + (- M3) = (- M2) + (0. (K,(len M1),(width M1))) by A7, MATRIX_3:5;
then ((- M1) + M3) + (- M3) = - M2 by A9, MATRIX_3:4;
then (- M1) + (M3 + (- M3)) = - M2 by A1, A2, A3, A4, A10, MATRIX_3:3;
then A11: (- M1) + (0. (K,(len M1),(width M1))) = - M2 by A7, MATRIX_3:5;
- M1 is Matrix of len M1, width M1,K by A6, A10, MATRIX_0:20;
then - M1 = - M2 by A11, MATRIX_3:4;
then - (- M1) = M2 by Th1;
hence M1 = M2 by Th1; :: thesis: verum

end;

suppose len M1 = 0 ; :: thesis: M1 = M2

hence M1 = M2 by A1, CARD_2:64; :: thesis: verum

end;

end;