
theorem Th10:
for I,J,K be closed_interval Subset of REAL,
 f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL
 st f is_continuous_on [:[:I,J:],K:] & f = g
 holds
  for e be Real st 0 < e ex r be Real st 0 < r
   & for x1,x2,y1,y2,z1,z2 be Real st x1 in I & x2 in I & y1 in J & y2 in J
      & z1 in K & z2 in K & |.x2-x1.| < r & |.y2-y1.| < r & |.z2-z1.| < r
        holds |. (|.g.|).([x2,y2,z2])-(|.g.|).([x1,y1,z1]) .| < e
proof
    let I,J,K be closed_interval Subset of REAL,
    f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL;
    assume that
A1:  f is_continuous_on [:[:I,J:],K:] and
A2:  f = g;
    reconsider f1 = ||.f.|| as
     PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real;
    ||.f.|| = |.g.| by A2,Th9;
    hence thesis by A1,Th8,MESFUN16:22;
end;
