reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th36:
for f be PartFunc of REAL i,REAL, x0 be Element of REAL i holds
  f is_continuous_in x0
    iff
 ( x0 in dom f &
   for r be Real st 0 < r
     ex s be Real st
       0 < s
     & for x be Element of REAL i st x in dom f & |. x-x0 .| < s
         holds |. f/.x - f/.x0 .| < r )
proof
   let f be PartFunc of REAL i,REAL, x0 be Element of REAL i;
   hereby assume f is_continuous_in x0; then
    consider y0 be Point of REAL-NS i,
             g be PartFunc of REAL-NS i,REAL such that
A1:  x0 = y0 & f = g & g is_continuous_in y0 by NFCONT_4:def 4;
    thus x0 in dom f by A1,NFCONT_1:8;
    let r be Real;
     assume 0 < r; then
     consider s be Real such that
A2:   0 < s
    & for y1 be Point of REAL-NS i st y1 in dom g & ||. y1- y0 .|| < s
        holds |. g/.y1-g/.y0 .|<r by A1,NFCONT_1:8;
     take s;
     thus 0<s by A2;
     let a be Element of REAL i;
     assume A3: a in dom f & |. a-x0 .| < s;
     reconsider y1=a as Point of REAL-NS i by REAL_NS1:def 4;
     ||. y1 -y0 .|| = |. a-x0 .| by A1,REAL_NS1:1,5;
     hence |. f/.a-f/.x0 .|<r by A1,A2,A3;
   end;
   assume
A4:  x0 in dom f
   & for r be Real st 0 < r
       ex s be Real st 0<s &
         for a be Element of REAL i st a in dom f & |. a-x0 .| < s
           holds |. f/.a-f/.x0 .| < r;
    reconsider y0=x0 as Point of REAL-NS i by REAL_NS1:def 4;
    reconsider g=f as PartFunc of REAL-NS i,REAL by REAL_NS1:def 4;
    now let r be Real;
     assume 0 < r; then
     consider s be Real such that
A5:   0 < s
    & for a be Element of REAL i
       st a in dom f & |. a-x0 .| < s holds |. f/.a-f/.x0 .| < r by A4;
     take s;
     thus 0 < s by A5;
     hereby let y1 be Point of REAL-NS i;
      assume A6: y1 in dom g & ||. y1- y0 .|| < s;
      reconsider a=y1 as Element  of REAL i by REAL_NS1:def 4;
      ||. y1 -y0 .|| = |. a-x0 .| by REAL_NS1:1,5;
      hence |. g/.y1-g/.y0 .| < r by A5,A6;
     end;
    end;
    hence f is_continuous_in x0 by NFCONT_4:def 4,A4,NFCONT_1:8;
   end;
