theorem Th45:
  for V,X being strict ComplexLinearSpace holds V is Subspace of X
  & X is Subspace of V implies V = X
proof
  let V,X be strict ComplexLinearSpace;
  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,Def8;
  then
A3: VV = VX;
  set AX = the addF of X;
  set AV = the addF of V;
  AV = AX||VV & AX = AV||VX by A1,A2,Def8;
  then
A4: AV = AX by A3,RELAT_1:72;
  set MX = the Mult of X;
  set MV = the Mult of V;
A5: MX = MV | [:COMPLEX,VX:] by A2,Def8;
  0.V = 0.X & MV = MX | [:COMPLEX,VV:] by A1,Def8;
  hence thesis by A3,A4,A5,RELAT_1:72;
end;
