reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

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;
