let K be Field; :: thesis: for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 holds
- (M1 + M2) = (- M1) + (- M2)

let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 implies - (M1 + M2) = (- M1) + (- M2) )
assume A1: ( len M1 = len M2 & width M1 = width M2 ) ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
A2: width (- M1) = width M1 by MATRIX_3:def 2;
then A3: width ((- M1) + (- M2)) = width M1 by MATRIX_3:def 3;
A4: ( len (M1 + M2) = len M1 & width (M1 + M2) = width M1 ) by MATRIX_3:def 3;
A5: len (- M1) = len M1 by MATRIX_3:def 2;
then A6: len ((- M1) + (- M2)) = len M1 by MATRIX_3:def 3;
A7: ( len (- M2) = len M2 & width (- M2) = width M2 ) by MATRIX_3:def 2;
per cases ( len M1 > 0 or len M1 = 0 ) by NAT_1:3;
suppose A8: len M1 > 0 ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
then A9: M2 is Matrix of len M1, width M1,K by ;
A10: M1 is Matrix of len M1, width M1,K by ;
(M1 + M2) + ((- M1) + (- M2)) = (M1 + M2) + ((- M2) + (- M1)) by
.= ((M1 + M2) + (- M2)) + (- M1) by
.= (M1 + (M2 + (- M2))) + (- M1) by
.= (M1 + (0. (K,(len M1),(width M1)))) + (- M1) by
.= M1 + (- M1) by
.= 0. (K,(len M1),(width M1)) by ;
hence - (M1 + M2) = (- M1) + (- M2) by A4, A6, A3, Th8; :: thesis: verum
end;
suppose A11: len M1 = 0 ; :: thesis: - (M1 + M2) = (- M1) + (- M2)
then len (- M1) = 0 by MATRIX_3:def 2;
then A12: len ((- M1) + (- M2)) = 0 by MATRIX_3:def 3;
len (M1 + M2) = 0 by ;
then len (- (M1 + M2)) = 0 by MATRIX_3:def 2;
hence - (M1 + M2) = (- M1) + (- M2) by ; :: thesis: verum
end;
end;