reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem
  for m being non zero Nat for f being Function of T,R^1 holds
  f = PROJ(m,m) * incl(f,m)
  proof
    let m be non zero Nat;
    let f be Function of T,R^1;
    let p be Point of T;
    set I = incl(f,m);
    reconsider G = m|->f.p as FinSequence of REAL;
    1 <= m by NAT_1:14;
    then
A1: m in Seg m;
A2: dom (m|->f.p) = Seg m;
    thus (PROJ(m,m)*I).p = PROJ(m,m).(I.p) by FUNCT_2:15
    .= I.p/.m by Def6
    .= G/.m by Def4
    .= G.m by A1,A2,PARTFUN1:def 6
    .= f.p by A1,FINSEQ_2:57;
  end;
