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

theorem Th45:
for f be PartFunc of REAL m,REAL st Z c= dom f holds
   f is_continuous_on Z
 iff
  for x0 be Element of REAL m, r be Real st x0 in Z & 0 < r
     ex s be Real st 0 < s
     & for x1 be Element of REAL m st x1 in Z & |. x1- x0 .| < s
          holds |. f/.x1 - f/.x0 .| < r
proof
   let f be PartFunc of REAL m,REAL;
   set g = <>*f;
   assume A1: Z c= dom f;
   hereby assume f is_continuous_on Z; then
A2: g is_continuous_on Z by A1,Th44;
    thus for x0 be Element of REAL m, r be Real st x0 in Z & 0<r
     ex s be Real
         st 0 <s & for x1 be Element of REAL m
         st x1 in Z & |. x1- x0 .| <s holds |. f/.x1-f/.x0 .|<r
    proof
     let x0 be Element of REAL m, r be Real;
     assume A3: x0 in Z & 0<r; then
     consider s be Real such that
A4:   0<s & for x1 be Element of REAL m st x1 in Z & |. x1- x0 .| < s
       holds |. g/.x1-g/.x0 .|<r by A2,PDIFF_7:38;
     take s;
     thus 0 < s by A4;
     hereby let x1 be Element of REAL m;
      assume A5: x1 in Z & |. x1- x0 .| < s; then
A6:   |. g/.x1-g/.x0 .|<r by A4;
      g/.x1 = <*f/.x1*> & g/.x0 = <*f/.x0*> by A5,A1,A3,Th6; then
      g/.x1-g/.x0 = <* f/.x1 - f/.x0 *> by RVSUM_1:29;
      hence |. f/.x1-f/.x0 .|<r by A6,Lm1;
     end;
    end;
   end;
   assume
A7: for x0 be Element of REAL m,
     r be Real st x0 in Z & 0<r ex s be Real
     st 0<s & for x1 be Element of REAL m
     st x1 in Z & |. x1- x0 .| <s holds |. f/.x1-f/.x0 .|<r;
A8:Z c= dom g by A1,Th3;
   for y0 be Element of REAL m, r be Real
     st y0 in Z & 0<r ex s be Real
    st 0 <s & for y1 be Element of REAL m
    st y1 in Z & |. y1- y0 .| < s
      holds |. g/.y1-g/.y0 .| < r
   proof
    let x0 be Element of REAL m,r be Real;
    assume A9: x0 in Z & 0<r; then
    consider s be Real such that
A10:  0<s & for x1 be Element of REAL m st x1 in Z & |. x1- x0 .| < s
       holds |. f/.x1-f/.x0 .|<r by A7;
    take s;
    thus 0 < s by A10;
    hereby let x1 be Element of REAL m;
     assume A11: x1 in Z & |. x1- x0 .| < s; then
A12: |. f/.x1-f/.x0 .|<r by A10;
     g/.x1 = <*f/.x1*> & g/.x0 = <*f/.x0*> by A1,A11,A9,Th6; then
     g/.x1-g/.x0 = <* f/.x1 - f/.x0 *> by RVSUM_1:29;
     hence |. g/.x1-g/.x0 .|<r by A12,Lm1;
    end;
   end; then
   g is_continuous_on Z by A8,PDIFF_7:38;
   hence f is_continuous_on Z by Th44;
end;
