
theorem Th23:
for I,J be closed_interval Subset of REAL,
  f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
  g be PartFunc of [:REAL,REAL:],REAL st f is_continuous_on [:I,J:] & f = g
holds
  for e be Real st 0 < e ex r be Real st 0 < r
   & for x1,x2,y1,y2 be Real st [x1,y1] in [:I,J:] & [x2,y2] in [:I,J:] &
      |.x2-x1.| < r & |.y2-y1.| < r holds
        |. (|.g.|).([x2,y2])-(|.g.|).([x1,y1]) .| < e
proof
    let I,J be closed_interval Subset of REAL,
    f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL;
    assume that
A1: f is_continuous_on [:I,J:] and
A2: f = g;

    ||.f.|| = |.g.| by A2,Th19;
    hence thesis by A1,Th22,Th18;
end;
