let V be RealLinearSpace; :: thesis: for V1 being Subset of V st V1 is linearly-closed & not V1 is empty holds
RLSStruct(# 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 RLSStruct(# V1,(Zero_ V1,V),(Add_ V1,V),(Mult_ V1,V) #) is Subspace of V )
assume A1: ( V1 is linearly-closed & not V1 is empty ) ; :: thesis: RLSStruct(# V1,(Zero_ V1,V),(Add_ V1,V),(Mult_ V1,V) #) is Subspace of V
A2: Zero_ V1,V = 0. V by A1, Def10;
A3: Add_ V1,V = the addF of V || V1 by A1, Def8;
Mult_ V1,V = the Mult of V | [:REAL ,V1:] by A1, Def9;
hence RLSStruct(# V1,(Zero_ V1,V),(Add_ V1,V),(Mult_ V1,V) #) is Subspace of V by A1, A2, A3, RLSUB_1:32; :: thesis: verum