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

theorem Th20:
  for K being Ring
  for A,B,C being Matrix of K st len B=len C & width B=width C &
  len B=width A & len B>0 & len A>0 holds A*(B-C) = A*B -A*C
proof
  let K be Ring;
  let A,B,C be Matrix of K;
  assume that
A1: len B=len C and
A2: width B=width C and
A3: len B=width A & len B>0 & len A>0;
A4: width (-C)=width C & len C=len (-C) by MATRIX_3:def 2;
  thus A*(B-C)=A*(B+ (- C)) by MATRIX_4:def 1
    .=A*B +A*(-C) by A1,A2,A3,A4,MATRIX_4:62
    .=A*B +-(A*C) by A1,A3,Th19
    .=A*B -(A*C) by MATRIX_4:def 1;
end;
