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

theorem Th37:
for f be PartFunc of REAL m,REAL, x0 be Element of REAL m holds
  f is_continuous_in x0 iff <>*f is_continuous_in x0
proof
   let f be PartFunc of REAL m,REAL, x0 be Element of REAL m;
   set g = <>*f;
   hereby assume A1: f is_continuous_in x0; then
A2: x0 in dom f by Th36; then
A3: x0 in dom g by Th3;
    now let r be Real;
     assume 0 < r; then
     consider s be Real such that
A4:   0 < s
    & for x1 be Element of REAL m st x1 in dom f & |. x1- x0 .| < s
        holds |. f/.x1-f/.x0 .| < r by A1,Th36;
     take s;
     thus 0 < s by A4;
     hereby let x1 be Element of REAL m;
      assume A5: x1 in dom g & |. x1- x0 .| < s; then
A6:   x1 in dom f by Th3; then
A7:   |. f/.x1-f/.x0 .|<r by A4,A5;
      g/.x1 = <* f/.x1 *> & g/.x0 = <* f/.x0 *> by A2,A6,Th6; then
      g/.x1-g/.x0 = <* f/.x1 - f/.x0 *> by RVSUM_1:29;
      hence |. g/.x1-g/.x0 .| < r by A7,Lm1;
     end;
    end;
    hence g is_continuous_in x0 by A3,PDIFF_7:36;
   end;
   assume A8:g is_continuous_in x0; then
    x0 in dom g by PDIFF_7:36; then
A9: x0 in dom f by Th3;
    now let r be Real;
     assume 0<r; then
     consider s be Real such that
A10:   0 < s
    & for x1 be Element of REAL m st x1 in dom g & |. x1- x0 .| < s
       holds |. g/.x1 - g/.x0 .| < r by A8,PDIFF_7:36;
     take s;
     thus 0<s by A10;
     hereby let x1 be Element of REAL m;
      assume A11: x1 in dom f & |. x1- x0 .| < s; then
      x1 in dom g by Th3; then
A12:   |. g/.x1 - g/.x0 .| < r by A10,A11;
      g/.x1 = <* f/.x1 *>  & g/.x0 = <* f/.x0 *> by A9,A11,Th6; then
      g/.x1-g/.x0 = <* f/.x1 - f/.x0 *> by RVSUM_1:29;
      hence |. f/.x1-f/.x0 .| < r by A12,Lm1;
     end;
    end;
    hence thesis by A9,Th36;
end;
