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

theorem Th49:
for f be PartFunc of REAL m,REAL, g be PartFunc of REAL-NS m,REAL
  st f=g holds Z c= dom f & f is_continuous_on Z iff g is_continuous_on Z
proof
   let f be PartFunc of REAL m,REAL, g be PartFunc of REAL-NS m,REAL;
   assume A1: f = g;
   hereby assume
A2: Z c= dom f;
    assume A3: f is_continuous_on Z;
    now let y0 be Point of REAL-NS m,r be Real;
     reconsider x0 = y0 as Element of REAL m by REAL_NS1:def 4;
     assume y0 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 |. f/.x1-f/.x0 .|<r by A2,A3,Th45;
     take s;
     thus 0 < s by A4;
     let y1 be Point of REAL-NS m;
     assume A5: y1 in Z & ||. y1- y0 .|| < s;
     reconsider x1=y1 as Element of REAL m by REAL_NS1:def 4;
     ||. y1- y0 .|| = |. x1- x0 .| by REAL_NS1:1,5;
     hence |. g/.y1-g/.y0 .|<r by A1,A5,A4;
    end;
    hence g is_continuous_on Z by A1,A2,NFCONT_1:20;
   end;
   assume A6: g is_continuous_on Z; then
A7:Z c= dom f by A1,NFCONT_1:20;
    now let x0 be Element of REAL m, r be Real;
     reconsider y0 = x0 as Point of  REAL-NS m by REAL_NS1:def 4;
     assume x0 in Z & 0 < r; then
     consider s be Real such that
A8:   0<s
    & for y1 be Point of REAL-NS m st y1 in Z & ||. y1- y0 .|| < s
        holds |. g/.y1-g/.y0 .|<r by A6,NFCONT_1:20;
     take s;
     thus 0 < s by A8;
     let x1 be Element of REAL m;
     assume A9: x1 in Z & |. x1- x0 .| < s;
     reconsider y1=x1 as Point of REAL-NS m by REAL_NS1:def 4;
     ||. y1- y0 .|| = |. x1- x0 .| by REAL_NS1:1,5;
     hence |. f/.x1-f/.x0 .|<r by A1,A9,A8;
    end;
    hence thesis by A7,Th45;
end;
