
theorem Th22:
  for V being RealUnitarySpace, W1,W2 being Subspace of V st the
  carrier of W1 c= the carrier of W2 holds W1 is Subspace of W2
proof
  let V be RealUnitarySpace;
  let W1,W2 be Subspace of V;
  set VW1 = the carrier of W1;
  set VW2 = the carrier of W2;
  set AV = the addF of V;
  set MV = the Mult of V;
  set SV = the scalar of V;
  assume
A1: the carrier of W1 c= the carrier of W2;
  then
A2: [:VW1,VW1:] c= [:VW2,VW2:] by ZFMISC_1:96;
  0.W1 = 0.V by Def1;
  hence the carrier of W1 c= the carrier of W2 & 0.W1 = 0.W2 by A1,Def1;
  the addF of W1 = AV||VW1 & the addF of W2 = AV||VW2 by Def1;
  hence the addF of W1 = (the addF of W2)||the carrier of W1 by A2,FUNCT_1:51;
A3: [:REAL,VW1:] c= [:REAL,VW2:] by A1,ZFMISC_1:95;
  the Mult of W1 = MV | [:REAL,VW1:] & the Mult of W2 = MV | [:REAL,VW2 :]
  by Def1;
  hence the Mult of W1 = (the Mult of W2) | [:REAL, the carrier of W1:] by A3,
FUNCT_1:51;
A4: [:VW1,VW1:] c= [:VW2,VW2:] by A1,ZFMISC_1:96;
  the scalar of W1 = SV||VW1 & the scalar of W2 = SV||VW2 by Def1;
  hence thesis by A4,FUNCT_1:51;
end;
