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

theorem Th34:
  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 * (v - w) = a * v - a * 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;
  thus a * (v - w) = a * v + a * (- w) by Def5
    .= a * v - a * w by Th25;
end;
