reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

theorem Th3:
  f is_continuous_in x0 iff x0 in dom f &
  for r st 0<r ex s st 0<s & for x1 st x1 in dom f & |.x1-x0.|<s
  holds |. f/.x1 - f/.x0 .|<r
proof
  hereby assume f is_continuous_in x0;
    then consider g be PartFunc of REAL,REAL-NS n
     such that
A1: f=g & g is_continuous_in x0;
    thus x0 in dom f by A1;
    thus
    for r st 0<r ex s st 0<s & for x1 st x1
    in dom f & |.x1-x0.|<s holds |. f/.x1 - f/.x0 .|<r
    proof
      let r;
      assume 0<r;
      then consider s such that
A2:   0<s & for x1 st x1
      in dom g & |.x1-x0.|<s holds ||. g/.x1 - g/.x0 .||<r by A1,NFCONT_3:8;
      take s;
      thus 0 < s by A2;
      let x1;
      assume x1 in dom f & |.x1-x0.|<s; then
A3:   ||. g/.x1 - g/.x0 .|| < r by A1,A2;
      g/.x1 = f/.x1 & g/.x0 = f/.x0 by A1,REAL_NS1:def 4;
      hence |. f/.x1 - f/.x0 .|<r by A3,REAL_NS1:1,5;
    end;
  end;
  assume
A4:x0 in dom f &
  for r st 0<r ex s st 0<s & for x1 st x1
    in dom f & |.x1-x0.|<s holds |. f/.x1 - f/.x0 .|<r;
  reconsider g= f as PartFunc of REAL,REAL-NS n
    by REAL_NS1:def 4;
  now let r be Real;
     reconsider rr=r as Real;
    assume 0<r;
    then consider s such that
A5: 0<s and
A6: for x1 st x1 in dom f & |.x1-x0.|<s holds |. f/.x1 - f/.x0 .|<rr by A4;
    take s;
    thus 0<s by A5;
    let x1;
    assume x1 in dom g & |.x1-x0.|<s; then
A7: |. f/.x1 - f/.x0 .|<r by A6;
    g/.x1 = f/.x1 & g/.x0 = f/.x0 by REAL_NS1:def 4;
    hence ||. g/.x1 - g/.x0 .||<r by A7,REAL_NS1:1,5;
  end;
  then g is_continuous_in x0 by A4,NFCONT_3:8;
  hence thesis;
end;
