reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th16:
  for K being Ring
  for A,B,C being Matrix of K st len B=len C & width B=width C &
  len A=width B & len B>0 & len A>0 holds (B-C)*A = B*A -C*A
proof
  let K be Ring;
  let A,B,C be Matrix of K;
  assume
A1: len B=len C & width B=width C & len A=width B & len B>0 & len A>0;
A2: width (-C)=width C & len C=len (-C) by MATRIX_3:def 2;
  thus (B-C)*A=(B+-C)*A by MATRIX_4:def 1
    .=B*A +(-C)*A by A1,A2,MATRIX_4:63
    .=B*A +-(C*A) by A1,Th15
    .=B*A -(C*A) by MATRIX_4:def 1;
end;
