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 Th45:
  for Ar be Subset of REAL-NS n,
      At be Subset of TOP-REAL n,
      v be Element of REAL-NS n,
      w be Element of TOP-REAL n
    st Ar = At & v = w & v in Affin Ar & Ar is affinely-independent
  holds
    v |-- Ar = w |-- At
  proof
    let Ar be Subset of REAL-NS n,
        At be Subset of TOP-REAL n,
        v be Element of REAL-NS n,
        w be Element of TOP-REAL n;
    assume
    A1: Ar = At & v=w & v in Affin Ar
      & Ar is affinely-independent;
    then
    A2: At is affinely-independent by Th41;
    reconsider h = v |-- Ar as Linear_Combination of At by A1,Th25;

    A3: Sum h
      = Sum (v |-- Ar) by Th23
     .= w by A1,RLAFFIN1:def 7;
    A4: sum h
      = sum (v |-- Ar) by Th44
     .= 1 by RLAFFIN1:def 7,A1;

    w in Affin At by A1,Th43;
    hence thesis by A2,A3,A4,RLAFFIN1:def 7;
  end;
