reserve x,y,y1,y2 for object;
reserve R for Ring;
reserve a for Scalar of R;
reserve V,X,Y for RightMod of R;
reserve u,u1,u2,v,v1,v2 for Vector of V;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Submodule of V;
reserve w,w1,w2 for Vector of W;

theorem Th25:
  for X,V being strict RightMod of R holds V is Submodule of X & X
  is Submodule of V implies V = X
proof
  let X,V be strict RightMod of R;
  assume that
A1: V is Submodule of X and
A2: X is Submodule 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 rmult of X;
  set MV = the rmult of V;
A5: MX = MV |([:VX,the carrier of R:] qua set) by A2,Def2;
  0.V = 0.X & MV = MX |([:VV,the carrier of R:] qua set) by A1,Def2;
  hence thesis by A3,A4,A5;
end;
