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 being Element of V holds
  v <> 0.V & a * v = b * v implies a = b
proof
  let V be add-associative right_zeroed right_complementable Abelian
  scalar-distributive scalar-associative scalar-unital vector-distributive
  non empty RLSStruct,
  v be Element of V;
  assume that
A1: v <> 0.V and
A2: a * v = b * v;
  0.V = a * v - b * v by A2,Def10
    .= (a - b) * v by Th35;
  then (- b) + a = 0 by A1,Th11;
  hence thesis;
end;
