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: Mult_ (V1,V) = the Mult of V | [:REAL,V1:] by A1, Def6;

( Zero_ (V1,V) = 0. V & Add_ (V1,V) = the addF of V || V1 ) by A1, Def5, Def7;

hence RLSStruct(# V1,(Zero_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)) #) is Subspace of V by A1, A2, RLSUB_1:24; :: thesis: verum

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: Mult_ (V1,V) = the Mult of V | [:REAL,V1:] by A1, Def6;

( Zero_ (V1,V) = 0. V & Add_ (V1,V) = the addF of V || V1 ) by A1, Def5, Def7;

hence RLSStruct(# V1,(Zero_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)) #) is Subspace of V by A1, A2, RLSUB_1:24; :: thesis: verum