let V be RealUnitarySpace; :: 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 )
set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
set AV = the addF of V;
set MV = the Mult of V;
set SV = the scalar of V;
assume A1: the carrier of W1 c= the carrier of W2 ; :: thesis: W1 is Subspace 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 Def1;
hence ( the carrier of W1 c= the carrier of W2 & 0. W1 = 0. W2 ) by A1, Def1; :: according to RUSUB_1:def 1 :: 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:] & the scalar of W1 = the scalar of W2 || 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 Def1;
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 | [:REAL, the carrier of W1:] & the scalar of W1 = the scalar of W2 || the carrier of W1 )
A3: [:REAL, the carrier of W1:] c= [:REAL, the carrier of W2:] by A1, ZFMISC_1:95;
( 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:] ) by Def1;
hence the Mult of W1 = the Mult of W2 | [:REAL, the carrier of W1:] by A3, FUNCT_1:51; :: thesis: the scalar of W1 = the scalar of W2 || the carrier of W1
A4: [: the carrier of W1, the carrier of W1:] c= [: the carrier of W2, the carrier of W2:] by A1, ZFMISC_1:96;
( the scalar of W1 = the scalar of V || the carrier of W1 & the scalar of W2 = the scalar of V || the carrier of W2 ) by Def1;
hence the scalar of W1 = the scalar of W2 || the carrier of W1 by A4, FUNCT_1:51; :: thesis: verum