
theorem Th21:
  for V,X,Y being RealUnitarySpace st V is Subspace of X & X is
  Subspace of Y holds V is Subspace of Y
proof
  let V,X,Y be RealUnitarySpace;
  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,Def1;
  hence the carrier of V c= the carrier of Y;
  0.V = 0.X by A1,Def1;
  hence 0.V = 0.Y by A2,Def1;
  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,Def1;
    then
A3: [:VV,VV:] c= [:VX,VX:] by ZFMISC_1:96;
    AV = AX||VV by A1,Def1;
    then AV = (AY||VX)||VV by A2,Def1;
    hence thesis by A3,FUNCT_1:51;
  end;
  thus the Mult of V = (the Mult of Y) | [:REAL, the carrier of V:]
  proof
    set MY = the Mult of Y;
    set VX = the carrier of X;
    set MX = the Mult of X;
    set VV = the carrier of V;
    set MV = the Mult of V;
    VV c= VX by A1,Def1;
    then
A4: [:REAL,VV:] c= [:REAL,VX:] by ZFMISC_1:95;
    MV = MX | [:REAL,VV:] by A1,Def1;
    then MV = (MY | [:REAL,VX:]) | [:REAL,VV:] by A2,Def1;
    hence thesis by A4,FUNCT_1:51;
  end;
  set SY = the scalar of Y;
  set SX = the scalar of X;
  set SV = the scalar of V;
  set VX = the carrier of X;
  set VV = the carrier of V;
  VV c= VX by A1,Def1;
  then
A5: [:VV,VV:] c= [:VX,VX:] by ZFMISC_1:96;
  SV = SX||VV by A1,Def1;
  then SV = SY||VX||VV by A2,Def1;
  hence thesis by A5,FUNCT_1:51;
end;
