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
M1 - M3 = (M1 - M2) - (M3 - 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 implies M1 - M3 = (M1 - M2) - (M3 - 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 ; :: thesis: M1 - M3 = (M1 - M2) - (M3 - M2)
A5: ( len (- M2) = len M2 & width (- M2) = width M2 ) by MATRIX_3:def 2;
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 - M3 = (M1 - M2) - (M3 - M2)

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

end;

suppose A11: len M1 = 0 ; :: thesis: M1 - M3 = (M1 - M2) - (M3 - M2)

then len (M1 - M2) = 0 by MATRIX_3:def 3;
then A12: len ((M1 - M2) - (M3 - M2)) = 0 by MATRIX_3:def 3;
len (M1 - M3) = 0 by A11, MATRIX_3:def 3;
hence M1 - M3 = (M1 - M2) - (M3 - M2) by A12, CARD_2:64; :: thesis: verum

end;

end;