let V be ComplexLinearSpace; :: thesis: 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
let V1 be Subset of V; :: thesis: ( V1 is linearly-closed & not V1 is empty implies CLSStruct(# V1,(Zero_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)) #) is Subspace of V )
assume that
A1: V1 is linearly-closed and
A2: not V1 is empty ; :: thesis: CLSStruct(# V1,(Zero_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)) #) is Subspace of V
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; :: thesis: verum