let V be 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
let V1 be Subset of V; ( 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
; CLSStruct(# V1,(Zero_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)) #) is Subspace of V
A3:
Add_ (V1,V) = the addF of V || V1
by A1, Def6;
A4:
Mult_ (V1,V) = the Mult of V | [:COMPLEX,V1:]
by A1, Def7;
Zero_ (V1,V) = 0. V
by A1, A2, Def8;
hence
CLSStruct(# V1,(Zero_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)) #) is Subspace of V
by A2, A3, A4, CLVECT_1:43; verum