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

theorem
  for x being FinSequence of REAL,A being Matrix of REAL st len x=len A
  & len x>0 holds x*(-A)=-(x*A)
proof
  let x be FinSequence of REAL,A be Matrix of REAL;
  assume that
A1: len x=len A and
A2: len x>0;
A3: width A=len (x*A) by A1,MATRIXR1:62;
A4: len A=len (-A) & width A=width (-A) by MATRIX_3:def 2;
  then
A5: len (x*(-A))=width A by A1,MATRIXR1:62;
  x*(-A)+(x*A)=x*(-A+A) by A1,A4,MATRIXR1:64
    .=x*(0_Rmatrix(len A,width A)) by Th31
    .=0*(width A) by A1,A2,MATRIXR1:66;
  hence thesis by A5,A3,Th1;
end;
