
theorem Th13:
  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
        st X is open & f = X --> d
    holds
      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.|
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;
  assume
A1: X is open & f = X --> d;
  let  x0 be Element of REAL m,r be Real;
  assume
A2: x0 in X & 0 < r;
  take s=1 qua Real;
  thus 0 < s;
  let x1 be Element of REAL m;
  assume
A3: x1 in X & |. x1- x0 .| < s;
  let v be Element of REAL m;
A4:diff(f,x1).v = (REAL m --> 0).v by A1,Th12,A3
               .= (0 qua Real) by FUNCOP_1:7;
   diff(f,x0).v = (REAL m --> 0).v by A1,Th12,A2
               .= (0 qua Real) by FUNCOP_1:7;
  hence |. diff(f,x1).v - diff(f,x0).v.| <= r * |.v.| by A4,A2,COMPLEX1:44;
end;
