reserve n for Nat,
  i for Integer,
  p, x, x0, y for Real,
  q for Rational,
  f for PartFunc of REAL,REAL;

theorem Th32:
  for f be PartFunc of REAL,REAL, Z be Subset of REAL, x be Real
  st x in Z for n be Nat holds f.x = Partial_Sums(Taylor(f,Z,x,x)).n
proof
  let f be PartFunc of REAL,REAL;
  let Z be Subset of REAL;
  let x be Real such that
A1: x in Z;
  defpred P[Nat] means f.x=Partial_Sums(Taylor(f, Z,x,x)).$1;
A2: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A3: P[k];
    thus Partial_Sums(Taylor(f, Z,x,x)).(k+1) =Partial_Sums(Taylor(f, Z,x,x)).
    k +Taylor(f, Z,x,x).(k+1) by SERIES_1:def 1
      .=f.x +(diff(f,Z).(k+1)).x * (x-x) |^ (k+1) /((k+1)!) by A3,Def7
      .=f.x + (diff(f,Z).(k+1)).x * (0|^k*0) /((k+1)!) by NEWTON:6
      .=f.x;
  end;
  Partial_Sums(Taylor(f, Z,x,x)).0 =Taylor(f, Z,x,x).0 by SERIES_1:def 1
    .=(diff(f,Z).0).x * (x-x) |^ 0 /(0!) by Def7
    .=(f|Z).x * (x-x) |^ 0 /(0!) by Def5
    .=(f|Z).x * 1/1 by NEWTON:4,12
    .=f.x by A1,FUNCT_1:49;
  then
A4: P[0];
  for k be Nat holds P[k] from NAT_1: sch 2(A4,A2);
  hence thesis;
end;
