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

theorem Th9:
for S,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 S,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 N01 be Neighbourhood of f/.x0;
    consider r such that
A2:  0<r and
A3:  { p where p is Point of S : ||. p - f/.x0 .|| < r } c= N01
        by NFCONT_1:def 1;
    set N1={ p where p is Point of S : ||. p - f/.x0 .|| < r };
    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,Th8;
    reconsider N = ].x0-s,x0+s.[ as Neighbourhood of x0 by A4,RCOMP_1:def 6;
    take N;
    let x1;
    assume that
A6:  x1 in dom f and
A7:  x1 in N;
    |.x1-x0.|<s by A7,RCOMP_1:1; then
    ||. f/.x1 - f/.x0 .||<r by A5,A6; then
    f/.x1 in N1;
    hence thesis by A3;
   end;
   assume
A8: 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;
   now let r;
    assume 0<r; then
    reconsider N1 = { p where p is Point of S : ||. p - f/.x0 .|| < r }
                 as Neighbourhood of f/.x0 by NFCONT_1:3;
    consider N2 being Neighbourhood of x0 such that
A9:  for x1 st x1 in dom f & x1 in N2 holds f/.x1 in N1 by A8;
    consider s be Real such that
A10: 0<s and
A11: N2 = ].x0-s,x0+s.[ by RCOMP_1:def 6;
    take s;
    for x1 st x1 in dom f & |.x1-x0.|<s holds ||. f/.x1 - f/.x0 .||<r
    proof
     let x1;
     assume that
A12:  x1 in dom f and
A13:  |.x1-x0.| < s;
     x1 in N2 by A11,A13,RCOMP_1:1; then
     f/.x1 in N1 by A9,A12; then
     ex p be Point of S st p=f/.x1 & ||. p - 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 A10;
   end;
   hence thesis by A8,Th8;
end;
