reserve V for RealLinearSpace;

theorem Th10:
  for W being Subspace of V for v being VECTOR of V, w being
  VECTOR of W st v = w holds Lin{w} = Lin{v}
proof
  let W be Subspace of V;
  let v be VECTOR of V, w be VECTOR of W such that
A1: v = w;
  reconsider W1 = Lin{w} as Subspace of V by RLSUB_1:27;
  now
    let u be VECTOR of V;
    hereby
      assume u in W1;
      then consider a being Real such that
A2:   u = a * w by RLVECT_4:8;
      u = a * v by A1,A2,RLSUB_1:14;
      hence u in Lin{v} by RLVECT_4:8;
    end;
    assume u in Lin{v};
    then consider a being Real such that
A3: u = a * v by RLVECT_4:8;
    u = a * w by A1,A3,RLSUB_1:14;
    hence u in W1 by RLVECT_4:8;
  end;
  hence thesis by RLSUB_1:31;
end;
