let V, X be strict ComplexLinearSpace; :: thesis: ( V is Subspace of X & X is Subspace of V implies V = X )
assume A1: ( V is Subspace of X & X is Subspace of V ) ; :: thesis: V = X
set VV = the carrier of V;
set VX = the carrier of X;
set AV = the addF of V;
set AX = the addF of X;
set MV = the Mult of V;
set MX = the Mult of X;
( the carrier of V c= the carrier of X & the carrier of X c= the carrier of V ) by A1, Def5;
then A2: the carrier of V = the carrier of X by XBOOLE_0:def 10;
A3: 0. V = 0. X by A1, Def5;
( the addF of V = the addF of X || the carrier of V & the addF of X = the addF of V || the carrier of X ) by A1, Def5;
then A4: the addF of V = the addF of X by A2, RELAT_1:101;
( the Mult of V = the Mult of X | [:COMPLEX ,the carrier of V:] & the Mult of X = the Mult of V | [:COMPLEX ,the carrier of X:] ) by A1, Def5;
hence V = X by A2, A3, A4, RELAT_1:101; :: thesis: verum