
theorem Th15:
  for m be non zero Element of NAT, f be PartFunc of REAL m,REAL,
      X be non empty Subset of REAL m, d be Real, i be Element of NAT
        st X is open & f = X --> d & 1 <= i & i <= m
    holds
      f is_partial_differentiable_on X,i & f`partial|(X,i) is_continuous_on X
proof
  let m be non zero Element of NAT, f be PartFunc of REAL m,REAL,
  X be non empty Subset of REAL m, d be Real, i be Element of NAT;
  assume
A1: X is open;
  assume
A2: f = X --> d;
  assume
A3: 1 <= i & i <= m;
A4:dom f = X by A2,FUNCT_2:def 1;
A5:f is_differentiable_on X by Th14,A2,A1;
  for x0 be Element of REAL m,r be Real st x0 in X & 0 < r
    ex s be Real
      st 0 < s & for x1 be Element of REAL m st x1 in X & |. x1- x0 .| < s
        holds for v be Element of REAL m
          holds |. diff(f,x1).v - diff(f,x0).v.| <= r * |.v.| by A2,Th13,A1;
  hence thesis by A3,A4,A1,A5,PDIFF_9:63;
end;
