let x be set ; :: thesis: for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of V st x in W1 & W1 is Subspace of W2 holds
x in W2
let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for V being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of V st x in W1 & W1 is Subspace of W2 holds
x in W2
let V be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for W1, W2 being Subspace of V st x in W1 & W1 is Subspace of W2 holds
x in W2
let W1, W2 be Subspace of V; :: thesis: ( x in W1 & W1 is Subspace of W2 implies x in W2 )
assume ( x in W1 & W1 is Subspace of W2 ) ; :: thesis: x in W2
then ( x in the carrier of W1 & the carrier of W1 c= the carrier of W2 ) by Def2, STRUCT_0:def 5;
hence x in W2 by STRUCT_0:def 5; :: thesis: verum