reserve x,y,y1,y2 for object;
reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr,
  V,X,Y for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF;
reserve a for Element of GF;
reserve u,u1,u2,v,v1,v2 for Element of V;
reserve W,W1,W2 for Subspace of V;
reserve V1 for Subset of V;
reserve w,w1,w2 for Element of W;

theorem Th25:
  for X,V being strict Abelian add-associative right_zeroed
  right_complementable vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty ModuleStr over GF holds V is
  Subspace of X & X is Subspace of V implies V = X
proof
  let X,V be strict Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
  non empty ModuleStr over GF;
  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;
  set MX = the lmult of X;
  set MV = the lmult of V;
A5: MX = MV |([:the carrier of GF,VX:] qua set) by A2,Def2;
  0.V = 0.X & MV = MX |([:the carrier of GF,VV:] qua set) by A1,Def2;
  hence thesis by A3,A4,A5;
end;
