let V, X, Y be RealLinearSpace; :: thesis: ( V is Subspace of X & X is Subspace of Y implies V is Subspace of Y )
assume A1: ( V is Subspace of X & X is Subspace of Y ) ; :: thesis: V is Subspace of Y
thus the carrier of V c= the carrier of Y :: according to RLSUB_1:def 2 :: thesis: ( 0. V = 0. Y & the addF of V = the addF of Y || the carrier of V & the Mult of V = the Mult of Y | [:REAL ,the carrier of V:] )

proof

( the carrier of V c= the carrier of X & the carrier of X c= the carrier of Y ) by A1, Def2;
hence the carrier of V c= the carrier of Y by XBOOLE_1:1; :: thesis: verum

end;

thus 0. V = 0. Y :: thesis: ( the addF of V = the addF of Y || the carrier of V & the Mult of V = the Mult of Y | [:REAL ,the carrier of V:] )

proof

( 0. V = 0. X & 0. X = 0. Y ) by A1, Def2;
hence 0. V = 0. Y ; :: thesis: verum

end;

thus the addF of V = the addF of Y || the carrier of V :: thesis: the Mult of V = the Mult of Y | [:REAL ,the carrier of V:]

proof

set AV = the addF of V;
set VV = the carrier of V;
set AX = the addF of X;
set VX = the carrier of X;
set AY = the addF of Y;
( the addF of V = the addF of X || the carrier of V & the addF of X = the addF of Y || the carrier of X & the carrier of V c= the carrier of X ) by A1, Def2;
then ( the addF of V = (the addF of Y || the carrier of X) || the carrier of V & [:the carrier of V,the carrier of V:] c= [:the carrier of X,the carrier of X:] ) by ZFMISC_1:119;
hence the addF of V = the addF of Y || the carrier of V by FUNCT_1:82; :: thesis: verum

end;

set MV = the Mult of V;
set VV = the carrier of V;
set MX = the Mult of X;
set VX = the carrier of X;
set MY = the Mult of Y;
( the Mult of V = the Mult of X | [:REAL ,the carrier of V:] & the Mult of X = the Mult of Y | [:REAL ,the carrier of X:] & the carrier of V c= the carrier of X ) by A1, Def2;
then ( the Mult of V = (the Mult of Y | [:REAL ,the carrier of X:]) | [:REAL ,the carrier of V:] & [:REAL ,the carrier of V:] c= [:REAL ,the carrier of X:] ) by ZFMISC_1:118;
hence the Mult of V = the Mult of Y | [:REAL ,the carrier of V:] by FUNCT_1:82; :: thesis: verum