let x be object ; :: thesis: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V st x in W1 & W1 is Submodule of W2 holds
x in W2
let R be Ring; :: thesis: for V being RightMod of R
for W1, W2 being Submodule of V st x in W1 & W1 is Submodule of W2 holds
x in W2
let V be RightMod of R; :: 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 Def2;
hence x in W2 ; :: thesis: verum