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 & M1 - M3 = M2 - M3 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 & M1 - M3 = M2 - M3 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: M1 - M3 = M2 - M3 ; :: thesis: M1 = M2
A6: ( len (- M3) = len M3 & width (- M3) = width M3 ) by MATRIX_3:def 2;

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

suppose A7: len M1 > 0 ; :: thesis: M1 = M2

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

end;

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

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

end;

end;