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 holds
(M3 - M1) - (M3 - M2) = M2 - M1
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 implies (M3 - M1) - (M3 - M2) = M2 - M1 )
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 ; :: thesis: (M3 - M1) - (M3 - M2) = M2 - M1

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

suppose A5: len M1 > 0 ; :: thesis: (M3 - M1) - (M3 - M2) = M2 - M1

then A6: M3 is Matrix of len M1, width M1,K by A1, A2, A3, A4, MATRIX_0:20;
A7: ( len (- M2) = len M2 & width (- M2) = width M2 ) by MATRIX_3:def 2;
A8: width (- M1) = width M1 by MATRIX_3:def 2;
then A9: width ((- M1) + M3) = width M1 by MATRIX_3:def 3;
A10: ( len (- M3) = len M3 & width (- M3) = width M3 ) by MATRIX_3:def 2;
A11: len (- M1) = len M1 by MATRIX_3:def 2;
then A12: len ((- M1) + M3) = len M1 by MATRIX_3:def 3;
A13: - M1 is Matrix of len M1, width M1,K by A5, A11, A8, MATRIX_0:20;
(M3 - M1) - (M3 - M2) = ((- M1) + M3) - (M3 + (- M2)) by A1, A2, A3, A4, A11, A8, MATRIX_3:2
.= ((- M1) + M3) + ((- M3) + (- (- M2))) by A2, A4, A7, Th12
.= ((- M1) + M3) + ((- M3) + M2) by Th1
.= (((- M1) + M3) + (- M3)) + M2 by A1, A2, A3, A4, A10, A12, A9, MATRIX_3:3
.= ((- M1) + (M3 + (- M3))) + M2 by A1, A2, A3, A4, A11, A8, MATRIX_3:3
.= ((- M1) + (0. (K,(len M1),(width M1)))) + M2 by A6, MATRIX_3:5
.= (- M1) + M2 by A13, MATRIX_3:4
.= M2 + (- M1) by A1, A3, A11, A8, MATRIX_3:2 ;
hence (M3 - M1) - (M3 - M2) = M2 - M1 ; :: thesis: verum

end;

suppose A14: len M1 = 0 ; :: thesis: (M3 - M1) - (M3 - M2) = M2 - M1

A15: len (M2 - M1) = len M2 by MATRIX_3:def 3;
len ((M3 - M1) - (M3 - M2)) = len (M3 - M1) by MATRIX_3:def 3
.= len M3 by MATRIX_3:def 3 ;
hence (M3 - M1) - (M3 - M2) = M2 - M1 by A1, A2, A14, A15, CARD_2:64; :: thesis: verum

end;

end;