reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem Th9:
  for f,x0 holds f is_continuous_in x0 iff x0 in dom f & for N1
being Neighbourhood of f/.x0 ex N being Neighbourhood of x0 st for x1 st x1 in
  dom f & x1 in N holds f/.x1 in N1
proof
  let f,x0;
  thus f is_continuous_in x0 implies x0 in dom f & for N1 being Neighbourhood
  of f/.x0 ex N being Neighbourhood of x0 st for x1 st x1 in dom f & x1 in N
  holds f/.x1 in N1
  proof
    assume
A1: f is_continuous_in x0;
    hence x0 in dom f;
    let N1 be Neighbourhood of f/.x0;
    consider r such that
A2: 0<r and
A3: {y where y is Point of T : ||.y-f/.x0 .|| < r} c= N1 by Def1;
    consider s such that
A4: 0<s and
A5: for x1 st x1 in dom f & ||. x1- x0 .|| <s holds ||. f/.x1- f/.x0
    .|| <r by A1,A2,Th7;
    reconsider s as Real;
    reconsider N={z where z is Point of S : ||.z-x0 .|| < s} as Neighbourhood
    of x0 by A4,Th3;
    take N;
    let x1;
    assume that
A6: x1 in dom f and
A7: x1 in N;
    ex z be Point of S st x1=z & ||.z-x0 .|| < s by A7;
    then ||. f/.x1- f/.x0 .|| <r by A5,A6;
    then f/.x1 in {y where y is Point of T : ||.y-f/.x0 .|| < r};
    hence thesis by A3;
  end;
  assume that
A8: x0 in dom f and
A9: for N1 being Neighbourhood of f/.x0 ex N being Neighbourhood of x0
  st for x1 st x1 in dom f & x1 in N holds f/.x1 in N1;
  now
    let r;
    assume
A10:    0<r;
    reconsider rr=r as Real;
    reconsider N1 = {y where y is Point of T : ||.y-f/.x0 .|| < rr} as
    Neighbourhood of f/.x0 by Th3,A10;
    consider N2 being Neighbourhood of x0 such that
A11: for x1 st x1 in dom f & x1 in N2 holds f/.x1 in N1 by A9;
    consider s such that
A12: 0<s and
A13: {z where z is Point of S : ||.z-x0 .|| < s} c= N2 by Def1;
    take s;
    for x1 st x1 in dom f & ||. x1- x0 .|| <s holds ||. f/.x1- f/.x0 .|| <r
    proof
      let x1;
      assume that
A14:  x1 in dom f and
A15:  ||. x1- x0 .|| <s;
      x1 in {z where z is Point of S : ||.z-x0 .|| < s} by A15;
      then f/.x1 in N1 by A11,A13,A14;
      then ex y be Point of T st f/.x1=y & ||.y-f/.x0 .|| < r;
      hence thesis;
    end;
    hence 0<s & for x1 st x1 in dom f & ||. x1- x0 .|| <s holds ||. f/.x1- f/.
    x0 .|| <r by A12;
  end;
  hence thesis by A8,Th7;
end;
