let K be Field; :: 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 ( len M1 = len M2 & width M1 = width M2 ) ; :: thesis: M1 = M1 - (M2 - M2)
then A1: M1 - (M2 - M2) = M1 - (0. (K,(len M1),(width M1))) by Th2
.= M1 + (0. (K,(len M1),(width M1))) by Th9 ;
per cases ( len M1 > 0 or len M1 = 0 ) by NAT_1:3;
suppose len M1 > 0 ; :: thesis: M1 = M1 - (M2 - M2)
then M1 is Matrix of len M1, width M1,K by MATRIX_0:20;
hence M1 = M1 - (M2 - M2) by A1, MATRIX_3:4; :: thesis: verum
end;
suppose A2: len M1 = 0 ; :: thesis: M1 = M1 - (M2 - M2)
then len (M1 - (M2 - M2)) = 0 by MATRIX_3:def 3;
hence M1 = M1 - (M2 - M2) by A2, CARD_2:64; :: thesis: verum
end;
end;