reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Subspace of V;
reserve x for set;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:COMPLEX,D:],D;

theorem Th46:
  V is Subspace of X & X is Subspace of Y implies V is Subspace of Y
proof
  assume that
A1: V is Subspace of X and
A2: X is Subspace of Y;
  the carrier of V c= the carrier of X & the carrier of X c= the carrier
  of Y by A1,A2,Def8;
  hence the carrier of V c= the carrier of Y;
  0.V = 0.X by A1,Def8;
  hence 0.V = 0.Y by A2,Def8;
  thus the addF of V = (the addF of Y)||the carrier of V
  proof
    set AY = the addF of Y;
    set VX = the carrier of X;
    set AX = the addF of X;
    set VV = the carrier of V;
    set AV = the addF of V;
    VV c= VX by A1,Def8;
    then
A3: [:VV,VV:] c= [:VX,VX:] by ZFMISC_1:96;
    AV = AX||VV by A1,Def8;
    then AV = AY||VX||VV by A2,Def8;
    hence thesis by A3,FUNCT_1:51;
  end;
  set MY = the Mult of Y;
  set MX = the Mult of X;
  set MV = the Mult of V;
  set VX = the carrier of X;
  set VV = the carrier of V;
  VV c= VX by A1,Def8;
  then
A4: [:COMPLEX,VV:] c= [:COMPLEX,VX:] by ZFMISC_1:95;
  MV = MX | [:COMPLEX,VV:] by A1,Def8;
  then MV = (MY | [:COMPLEX,VX:]) | [:COMPLEX,VV:] by A2,Def8;
  hence thesis by A4,FUNCT_1:51;
end;
