
theorem Th20:
  for V,X being strict RealUnitarySpace holds V is Subspace of X &
  X is Subspace of V implies V = X
proof
  let V,X be strict RealUnitarySpace;
  assume that
A1: V is Subspace of X and
A2: X is Subspace of V;
  set VX = the carrier of X;
  set VV = the carrier of V;
  VV c= VX & VX c= VV by A1,A2,Def1;
  then
A3: VV = VX;
  set MX = the Mult of X;
  set MV = the Mult of V;
  MV = MX | [:REAL,VV:] & MX = MV | [:REAL,VX:] by A1,A2,Def1;
  then
A4: MV = MX by A3,RELAT_1:72;
  set AX = the addF of X;
  set AV = the addF of V;
  AV = AX||VV & AX = AV||VX by A1,A2,Def1;
  then
A5: AV = AX by A3,RELAT_1:72;
  set SX = the scalar of X;
  set SV = the scalar of V;
A6: SX = SV||VX by A2,Def1;
  0.V = 0.X & SV = SX||VV by A1,Def1;
  hence thesis by A3,A5,A4,A6,RELAT_1:72;
end;
