let V be Z_Module; :: 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 AV = the addF of V;
set MV = the Mult of V;
assume A1: the carrier of W1 c= the carrier of W2 ; :: thesis: W1 is Submodule of W2
then A2: [: the carrier of W1, the carrier of W1:] c= [: the carrier of W2, the carrier of W2:] by ZFMISC_1:96;
0. W1 = 0. V by Def9;
hence ( the carrier of W1 c= the carrier of W2 & 0. W1 = 0. W2 ) by A1, Def9; :: according to ZMODUL01:def 9 :: thesis: ( the addF of W1 = the addF of W2 || the carrier of W1 & the Mult of W1 = the Mult of W2 | [:INT, the carrier of W1:] )
( 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 Def9;
hence the addF of W1 = the addF of W2 || the carrier of W1 by A2, FUNCT_1:51; :: thesis: the Mult of W1 = the Mult of W2 | [:INT, the carrier of W1:]
A3: [:INT, the carrier of W1:] c= [:INT, the carrier of W2:] by A1, ZFMISC_1:95;
( the Mult of W1 = the Mult of V | [:INT, the carrier of W1:] & the Mult of W2 = the Mult of V | [:INT, the carrier of W2:] ) by Def9;
hence the Mult of W1 = the Mult of W2 | [:INT, the carrier of W1:] by A3, FUNCT_1:51; :: thesis: verum