reserve V for non empty RLSStruct;
reserve x,y,y1 for set;
reserve v for VECTOR of V;
reserve a,b for Real;

theorem
  for V being add-associative right_zeroed right_complementable Abelian
  scalar-distributive scalar-associative scalar-unital vector-distributive
  non empty RLSStruct,
  v,w being Element of V holds
  a <> 0 & a * v = a * w implies v = w
proof
  let V be add-associative right_zeroed right_complementable Abelian
  scalar-distributive scalar-associative scalar-unital vector-distributive
  non empty RLSStruct,
  v,w be Element of V;
  assume that
A1: a <> 0 and
A2: a * v = a * w;
  0.V = a * v - a * w by A2,Def10
    .= a * (v - w) by Th34;
  then v - w = 0.V by A1,Th11;
  hence thesis by Th21;
end;
