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)
then A2: ( len (- M2) = len M1 & width (- M2) = width M1 ) by MATRIX_3:def 2;
A3: ( len (- M1) = len M1 & width (- M1) = width M1 ) by MATRIX_3:def 2;
A4: ( len (M1 + M2) = len M1 & width (M1 + M2) = width M1 ) by MATRIX_3:def 3;
(- M2) - ((- M1) - M2) = (- M2) + ((- (- M1)) + (- (- M2))) by A2, A3, Th12
.= (- M2) + (M1 + (- (- M2))) by Th1
.= (- M2) + (M1 + M2) by Th1
.= (M1 + M2) + (- M2) by A2, A4, MATRIX_3:4 ;
hence M1 = (- M2) - ((- M1) - M2) by A1, Th38; :: thesis: verum