reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem
  for n, m be Nat
  for M be Matrix of n,m,F_Real
  for A be affinely-independent Subset of REAL-NS n
    st the_rank_of M = n
  holds
    for v be Element of REAL-NS n
      st v in Affin A
    holds
      (Mx2Tran M).v in Affin((Mx2Tran M) .: A)
    & (for f be n -element real-valued FinSequence
       holds (v |-- A).f = (((Mx2Tran M).v)
          |-- ((Mx2Tran M) .: A)) . ((Mx2Tran M).f))
  proof
    let n, m be Nat;
    let M be Matrix of n,m,F_Real;
    let B be affinely-independent Subset of REAL-NS n;
    assume
    A1: the_rank_of M = n;

    let w be Element of REAL-NS n;
    assume
    A2: w in Affin B;

    reconsider A = B as affinely-independent Subset of TOP-REAL n by Th41;
    reconsider v = w as Element of TOP-REAL n by Th4;
    A3: v in Affin A by A2,Th43;
    v |-- A = w |-- B by A2,Th45;
    hence thesis by A1,A3,MATRTOP2:25;
  end;
