let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for W1, W2 being Subspace of V st the carrier of W1 c= the carrier of W2 holds

W1 is Subspace of W2

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for W1, W2 being Subspace of V st the carrier of W1 c= the carrier of W2 holds

W1 is Subspace of W2

let W1, W2 be Subspace of V; :: thesis: ( the carrier of W1 c= the carrier of W2 implies W1 is Subspace of W2 )

set VW1 = the carrier of W1;

set VW2 = the carrier of W2;

set MW1 = the lmult of W1;

set MW2 = the lmult of W2;

set AV = the addF of V;

set MV = the lmult of V;

A1: ( the addF of W1 = the addF of V || the carrier of W1 & the addF of W2 = the addF of V || the carrier of W2 ) by Def2;

assume A2: the carrier of W1 c= the carrier of W2 ; :: thesis: W1 is Subspace of W2

then [: the carrier of W1, the carrier of W1:] c= [: the carrier of W2, the carrier of W2:] by ZFMISC_1:96;

then A3: the addF of W1 = the addF of W2 || the carrier of W1 by A1, FUNCT_1:51;

A4: ( the lmult of W1 = the lmult of V | [: the carrier of GF, the carrier of W1:] & the lmult of W2 = the lmult of V | [: the carrier of GF, the carrier of W2:] ) by Def2;

[: the carrier of GF, the carrier of W1:] c= [: the carrier of GF, the carrier of W2:] by A2, ZFMISC_1:95;

then A5: the lmult of W1 = the lmult of W2 | [: the carrier of GF, the carrier of W1:] by A4, FUNCT_1:51;

( 0. W1 = 0. V & 0. W2 = 0. V ) by Def2;

hence W1 is Subspace of W2 by A2, A3, A5, Def2; :: thesis: verum

for W1, W2 being Subspace of V st the carrier of W1 c= the carrier of W2 holds

W1 is Subspace of W2

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for W1, W2 being Subspace of V st the carrier of W1 c= the carrier of W2 holds

W1 is Subspace of W2

let W1, W2 be Subspace of V; :: thesis: ( the carrier of W1 c= the carrier of W2 implies W1 is Subspace of W2 )

set VW1 = the carrier of W1;

set VW2 = the carrier of W2;

set MW1 = the lmult of W1;

set MW2 = the lmult of W2;

set AV = the addF of V;

set MV = the lmult of V;

A1: ( the addF of W1 = the addF of V || the carrier of W1 & the addF of W2 = the addF of V || the carrier of W2 ) by Def2;

assume A2: the carrier of W1 c= the carrier of W2 ; :: thesis: W1 is Subspace of W2

then [: the carrier of W1, the carrier of W1:] c= [: the carrier of W2, the carrier of W2:] by ZFMISC_1:96;

then A3: the addF of W1 = the addF of W2 || the carrier of W1 by A1, FUNCT_1:51;

A4: ( the lmult of W1 = the lmult of V | [: the carrier of GF, the carrier of W1:] & the lmult of W2 = the lmult of V | [: the carrier of GF, the carrier of W2:] ) by Def2;

[: the carrier of GF, the carrier of W1:] c= [: the carrier of GF, the carrier of W2:] by A2, ZFMISC_1:95;

then A5: the lmult of W1 = the lmult of W2 | [: the carrier of GF, the carrier of W1:] by A4, FUNCT_1:51;

( 0. W1 = 0. V & 0. W2 = 0. V ) by Def2;

hence W1 is Subspace of W2 by A2, A3, A5, Def2; :: thesis: verum