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

theorem Th11:
  for V being add-associative right_zeroed right_complementable
  scalar-distributive scalar-associative scalar-unital vector-distributive
  non empty RLSStruct,
  v being Element of V holds
  a * v = 0.V implies a = 0 or v = 0.V
proof
  let V be add-associative right_zeroed right_complementable
  scalar-distributive scalar-associative scalar-unital vector-distributive
  non empty RLSStruct,
  v be Element of V;
  assume that
A1: a * v = 0.V and
A2: a <> 0;
  thus v = 1 * v by Def8
    .= (a" * a) * v by A2,XCMPLX_0:def 7
    .= a" * 0.V by A1,Def7
    .= 0.V;
end;
