theorem Th26:
  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,Def2;
  then
A3: the carrier of V c= the carrier of Y;
A4: 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,Def2;
    then
A5: [:VV,VV:] c= [:VX,VX:] by ZFMISC_1:96;
    AV = AX||VV by A1,Def2;
    then AV = AY||VX||VV by A2,Def2;
    hence thesis by A5,FUNCT_1:51;
  end;
  set MY = the lmult of Y;
  set MX = the lmult of X;
  set MV = the lmult of V;
  set VX = the carrier of X;
  set VV = the carrier of V;
  VV c= VX by A1,Def2;
  then
A6: [:the carrier of GF,VV:] c= [:the carrier of GF,the carrier of X:] by
ZFMISC_1:95;
  MV = MX |([:the carrier of GF,VV:] qua set) by A1,Def2;
  then MV = MY |([:the carrier of GF,VX:] qua set) |([:the carrier of GF,VV:]
  qua set) by A2,Def2;
  then
A7: MV = MY |([:the carrier of GF,VV:] qua set) by A6,FUNCT_1:51;
  0.V = 0.X by A1,Def2;
  then 0.V = 0.Y by A2,Def2;
  hence thesis by A3,A4,A7,Def2;
end;
