reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:REAL,D:],D;

theorem Th26:
  for V,X being strict RealLinearSpace holds V is Subspace of X &
  X is Subspace of V implies V = X
proof
  let V,X be strict RealLinearSpace;
  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,Def2;
  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,Def2;
  then
A4: AV = AX by A3,RELAT_1:72;
  set MX = the Mult of X;
  set MV = the Mult of V;
A5: MX = MV | [:REAL,VX:] by A2,Def2;
  0.V = 0.X & MV = MX | [:REAL,VV:] by A1,Def2;
  hence thesis by A3,A4,A5,RELAT_1:72;
end;
