let K be Ring; :: 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 A1, MATRIX_0:20;
A10: M1 is Matrix of len M1, width M1,K by A8, MATRIX_0:20;
(M1 + M2) + ((- M1) + (- M2)) = (M1 + M2) + ((- M2) + (- M1)) by A1, A5, A2, A7, MATRIX_3:2
.= ((M1 + M2) + (- M2)) + (- M1) by A1, A7, A4, MATRIX_3:3
.= (M1 + (M2 + (- M2))) + (- M1) by A1, MATRIX_3:3
.= (M1 + (0. (K,(len M1),(width M1)))) + (- M1) by A9, MATRIX_3:5
.= M1 + (- M1) by A10, MATRIX_3:4
.= 0. (K,(len M1),(width M1)) by A10, MATRIX_3:5 ;
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 A11, MATRIX_3:def 3;
then len (- (M1 + M2)) = 0 by MATRIX_3:def 2;
hence - (M1 + M2) = (- M1) + (- M2) by A12, CARD_2:64; :: thesis: verum

end;

end;