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 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 x > 0;
A3: len ColVec2Mx x=len x by A2,MATRIXR1:def 9;
  width ColVec2Mx x=1 by A2,MATRIXR1:def 9;
  then
A4: 1<=width(A*(ColVec2Mx x)) by A1,A3,MATRIX_3:def 4;
  thus A*(-x) =Col(A*((-1)*ColVec2Mx x),1) by A2,MATRIXR1:47
    .=Col((-1)*(A*(ColVec2Mx x)),1) by A1,A3,MATRIXR1:40
    .=-(A*x) by A4,MATRIXR1:56;
end;
