
theorem Th40: :: CSSPACE:13
  for V being ComplexLinearSpace
  for V1 being Subset of V st V1 is linearly-closed & not V1 is empty holds
  CLSStruct(# V1,(Zero_(V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)) #) is Subspace of V
proof
  let V be ComplexLinearSpace;
  let V1 be Subset of V;
  assume that
A1: V1 is linearly-closed and
A2: not V1 is empty;
  for v, u being VECTOR of V st v in V1 & u in V1 holds
     v + u in V1 by A1;
  then V1 is add-closed by IDEAL_1:def 1;
  then
A3: Add_ (V1,V) = (the addF of V)|| V1  by A2,C0SP1:def 5;
A4:Mult_ (V1,V) = (the Mult of V) | [:COMPLEX,V1:] by A1,CSSPACE:def 9;
  Zero_ (V1,V) = 0. V by A1,A2,CSSPACE:def 10;
  hence CLSStruct(# V1,(Zero_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)) #)
            is Subspace of V by A2,A3,A4,CLVECT_1:43;
end;
