let V be RealLinearSpace; :: thesis: for W1, W2 being Subspace of V st the carrier of W1 c= the carrier of W2 holds
W1 is Subspace of W2
let W1, W2 be Subspace of V; :: thesis: ( the carrier of W1 c= the carrier of W2 implies W1 is Subspace of W2 )
assume A1: the carrier of W1 c= the carrier of W2 ; :: thesis: W1 is Subspace 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;
( 0. W1 = 0. V & 0. W2 = 0. V ) by Def2;
hence ( the carrier of W1 c= the carrier of W2 & 0. W1 = 0. W2 ) by A1; :: according to RLSUB_1:def 2 :: thesis: ( the addF of W1 = the addF of W2 || the carrier of W1 & the Mult of W1 = the Mult of W2 | [:REAL ,the carrier of W1:] )
thus the addF of W1 = the addF of W2 || the carrier of W1 :: thesis: the Mult of W1 = the Mult of W2 | [:REAL ,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 by FUNCT_1:82; :: thesis: verum

end;

( the Mult of W1 = the Mult of V | [:REAL ,the carrier of W1:] & the Mult of W2 = the Mult of V | [:REAL ,the carrier of W2:] & [:REAL ,the carrier of W1:] c= [:REAL ,the carrier of W2:] ) by A1, Def2, ZFMISC_1:118;
hence the Mult of W1 = the Mult of W2 | [:REAL ,the carrier of W1:] by FUNCT_1:82; :: thesis: verum