let V be Z_Module; :: thesis: for x being set
for W1, W2 being Submodule of V st x in W1 & W1 is Submodule of W2 holds
x in W2
let x be set ; :: thesis: for W1, W2 being Submodule of V st x in W1 & W1 is Submodule of W2 holds
x in W2
let W1, W2 be Submodule of V; :: thesis: ( x in W1 & W1 is Submodule of W2 implies x in W2 )
assume ( x in W1 & W1 is Submodule of W2 ) ; :: thesis: x in W2
then ( x in the carrier of W1 & the carrier of W1 c= the carrier of W2 ) by Def9;
hence x in W2 ; :: thesis: verum