let K be Ring; :: thesis: for M1, M2 being Matrix of K st len M1 = len M2 & width M1 = width M2 holds
M1 = (M1 - M2) + M2
let M1, M2 be Matrix of K; :: thesis: ( len M1 = len M2 & width M1 = width M2 implies M1 = (M1 - M2) + M2 )
assume A1: ( len M1 = len M2 & width M1 = width M2 ) ; :: thesis: M1 = (M1 - M2) + M2
then A2: ( len (- M2) = len M1 & width (- M2) = width M1 ) by MATRIX_3:def 2;
hence (M1 - M2) + M2 = M1 + ((- M2) + M2) by MATRIX_3:3
.= M1 + (M2 - M2) by A1, A2, MATRIX_3:2
.= M1 by A1, Th20 ;
:: thesis: verum