
theorem Th6:
  for F being Field for V being VectSp of F for W1,W2,W3,W4 being
  Subspace of V st W1 is Subspace of W2 & W3 is Subspace of W4 & (Omega).W1=
  (Omega).W3 & (Omega).W2=(Omega).W4 holds segment(W1,W2) = segment(W3,W4)
proof
  let F be Field;
  let V be VectSp of F;
  let W1,W2,W3,W4 be Subspace of V;
  assume that
A1: W1 is Subspace of W2 and
A2: W3 is Subspace of W4 and
A3: (Omega).W1=(Omega).W3 and
A4: (Omega).W2=(Omega).W4;
  thus segment(W1,W2) c= segment(W3,W4)
  proof
    let a be object;
    assume
A5: a in segment(W1,W2);
    then ex A1 being strict Subspace of V st A1=a by VECTSP_5:def 3;
    then reconsider A=a as strict Subspace of V;
    A is Subspace of W2 by A1,A5,Def1;
    then A is Subspace of (Omega).W2 by Th2;
    then
A6: A is Subspace of W4 by A4,Th3;
    W1 is Subspace of A by A1,A5,Def1;
    then (Omega).W1 is Subspace of A by Th4;
    then W3 is Subspace of A by A3,Th5;
    hence thesis by A2,A6,Def1;
  end;
  let a be object;
  assume
A7: a in segment(W3,W4);
  then ex A1 being strict Subspace of V st A1=a by VECTSP_5:def 3;
  then reconsider A=a as strict Subspace of V;
  A is Subspace of W4 by A2,A7,Def1;
  then A is Subspace of (Omega).W4 by Th2;
  then
A8: A is Subspace of W2 by A4,Th3;
  W3 is Subspace of A by A2,A7,Def1;
  then (Omega).W3 is Subspace of A by Th4;
  then W1 is Subspace of A by A3,Th5;
  hence thesis by A1,A8,Def1;
end;
