reserve a,b,r for Real;

theorem Th5:
  for V be non empty RLSStruct, V1 be add-closed multi-closed non
empty Subset of V,
   a be Real, v be VECTOR of V, w be VECTOR of RLSStruct (# V1,
    In (0.V, V1), add|(V1,V), Mult_ V1 #) st w = v holds a*w = a*v
proof
  let V be non empty RLSStruct, V1 be add-closed multi-closed non empty Subset
of V, a be Real,
   v be VECTOR of V, w be VECTOR of RLSStruct (# V1,In (0.V, V1),
    add|(V1,V),Mult_ V1 #);
   reconsider a as Element of REAL by XREAL_0:def 1;
  assume
A1: w = v;
  then [a,v] in [:REAL,V1:] by ZFMISC_1:87;
  hence thesis by A1,FUNCT_1:49;
end;
