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 A=len x
  & len x>0 & width A>0 holds (-x)*A=-(x*A)
proof
  let x be FinSequence of REAL,A be Matrix of REAL;
  assume that
A1: len A=len x and
A2: len x>0 and
A3: width A>0;
A4: (A@)*x=x*A by A1,A2,A3,MATRIXR1:52;
A5: width (A@)=len x by A1,A3,MATRIX_0:54;
  then
A6: (A@)*(-x)=(-1)*((A@)*x) by A2,MATRIXR1:59;
A7: len (-x)=len x by RVSUM_1:114;
  len (A@)>0 by A3,MATRIX_0:54;
  then (-x)*((A@)@)=(A@)*(-x) by A2,A5,A7,MATRIXR1:53;
  hence thesis by A1,A2,A3,A6,A4,MATRIX_0:57;
end;
