theorem
  for B1 be FinSequence of V1,W1 be Subspace of V1, B2 be FinSequence of
  W1 st B1 = B2 holds lmlt(p,B1) = lmlt(p,B2)
proof
  let B1 be FinSequence of V1,W1 be Subspace of V1, B2 be FinSequence of W1
  such that
A1: B1 = B2;
  set M2=lmlt(p,B2);
  set M1=lmlt(p,B1);
A2: dom M1=dom p/\dom B1 by Lm1;
A3: dom M2=dom p/\dom B2 by Lm1;
  now
    let i such that
A4: i in dom M1;
    i in dom p by A2,A4,XBOOLE_0:def 4;
    then
A5: p.i=p/.i by PARTFUN1:def 6;
A6: i in dom B1 by A2,A4,XBOOLE_0:def 4;
    then
A7: B2.i=B2/.i by A1,PARTFUN1:def 6;
A8: B1.i=B1/.i by A6,PARTFUN1:def 6;
    hence M1.i = p/.i * B1/.i by A4,A5,FUNCOP_1:22
      .= p/.i * B2/.i by A1,A6,A8,PARTFUN1:def 6,VECTSP_4:14
      .= M2.i by A1,A2,A3,A4,A5,A7,FUNCOP_1:22;
  end;
  hence thesis by A1,A3,Lm1;
end;
