let K be Field; :: 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 ;
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 ;
(M1 - M2) - (M3 - M2) = (M1 + (- M2)) + ((- M3) + (- (- M2))) by A2, A4, A5, Th12
.= (M1 + (- M2)) + ((- M3) + M2) by Th1
.= (M1 + (- M2)) + (M2 + (- M3)) by
.= ((M1 + (- M2)) + M2) + (- M3) by
.= (M1 + ((- M2) + M2)) + (- M3) by
.= (M1 + (M2 + (- M2))) + (- M3) by
.= (M1 + (0. (K,(len M1),(width M1)))) + (- M3) by
.= M1 + (- M3) by ;
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 ;
hence M1 - M3 = (M1 - M2) - (M3 - M2) by ; :: thesis: verum
end;
end;