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 )
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;
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 Submodule 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, the carrier of W1:] by A1, FUNCT_1:51;
A4: ( 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:] ) by Def2;
[: the carrier of W1, the carrier of R:] c= [: the carrier of W2, the carrier of R:] by A2, ZFMISC_1:95;
then A5: the rmult of W1 = the rmult of W2 | [: the carrier of W1, the carrier of R:] by A4, FUNCT_1:51;
( 0. W1 = 0. V & 0. W2 = 0. V ) by Def2;
hence W1 is Submodule of W2 by A2, A3, A5, Def2; :: thesis: verum