let R be Ring; :: thesis: for V being RightMod of R
for W1, W2 being Submodule of V st the carrier of W1 c= the carrier of W2 holds
W1 is Submodule of W2
let V be RightMod of R; :: thesis: for W1, W2 being Submodule of V st the carrier of W1 c= the carrier of W2 holds
W1 is Submodule of W2
let W1, W2 be Submodule of V; :: thesis: ( the carrier of W1 c= the carrier of W2 implies W1 is Submodule of W2 )
assume A1: the carrier of W1 c= the carrier of W2 ; :: thesis: W1 is Submodule of W2
set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
set MW1 = the rmult of W1;
set MW2 = the rmult of W2;
set AV = the addF of V;
set MV = the rmult of V;
A2: ( 0. W1 = 0. V & 0. W2 = 0. V ) by Def2;
A3: the addF of W1 = the addF of W2 | [:the carrier of W1,the carrier of W1:]

proof

( the addF of W1 = the addF of V || the carrier of W1 & the addF of W2 = the addF of V || the carrier of W2 & [:the carrier of W1,the carrier of W1:] c= [:the carrier of W2,the carrier of W2:] ) by A1, Def2, ZFMISC_1:119;
hence the addF of W1 = the addF of W2 | [:the carrier of W1,the carrier of W1:] by FUNCT_1:82; :: thesis: verum

end;

( the rmult of W1 = the rmult of V | [:the carrier of W1,the carrier of R:] & the rmult of W2 = the rmult of V | [:the carrier of W2,the carrier of R:] & [:the carrier of W1,the carrier of R:] c= [:the carrier of W2,the carrier of R:] ) by A1, Def2, ZFMISC_1:118;
then the rmult of W1 = the rmult of W2 | [:the carrier of W1,the carrier of R:] by FUNCT_1:82;
hence W1 is Submodule of W2 by A1, A2, A3, Def2; :: thesis: verum