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

theorem Th19:
  for K being Ring
  for A,C being Matrix of K st width A=len C & len C>0
  holds A*(-C) = -(A*C)
proof
  let K be Ring;
  let A,C be Matrix of K;
  assume that
A1: width A=len C and
A3: len C>0;
A4: len C=len (-C) by MATRIX_3:def 2;
  then
A5: len A=len (A*(-C)) by A1,MATRIX_3:def 4;
  width (-C)=width C by MATRIX_3:def 2;
  then
A6: width (A*(-C))=width C by A1,A4,MATRIX_3:def 4;
  reconsider D=C as Matrix of (len C),(width C),K by A3,MATRIX_0:20;
A7: len (A*C)=len A & width (A*C)=width C by A1,MATRIX_3:def 4;
  len C = len (-C) & width (-C)=width C by MATRIX_3:def 2;
  then A*C +(A*(-C)) =A*(D+-D) by A1,MATRIX_4:62
    .= A*(0.(K,len C,width C)) by MATRIX_3:5
    .= 0.(K,len A,width C) by A1,A3,Th18;
  hence thesis by A7,A5,A6,MATRIX_4:8;
end;
