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=width
  A & len A>0 & len x> 0 holds (-A)*x=-(A*x)
proof
  let x be FinSequence of REAL,A be Matrix of REAL;
  assume that
A1: len x=width A and
A2: len A>0 and
A3: len x> 0;
A4: len ColVec2Mx x=len x & width ColVec2Mx x=1 by A3,MATRIXR1:def 9;
  then
A5: 1<=width(A*(ColVec2Mx x)) by A1,MATRIX_3:def 4;
  thus (-A)*x =Col(((-1)*A*(ColVec2Mx x)),1) by Th9
    .=Col((-1)*(A*(ColVec2Mx x)),1) by A1,A2,A3,A4,MATRIXR1:41
    .=-(A*x) by A5,MATRIXR1:56;
end;
