
theorem Th8:
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;

    set E = [:[:I,J:],K:];
A3: f is_uniformly_continuous_on E by A1,Th2,NFCONT_2:10;

    let e be Real;
    assume 0 < e; then
    consider r be Real such that
A4: 0 < r
  & for x1,x2,y1,y2,z1,z2 be Real st [x1,y1,z1] in E & [x2,y2,z2] in E
    & |.x2-x1.| < r & |.y2-y1.| < r & |.z2-z1.| < r holds
       |. g.([x2,y2,z2])-g.([x1,y1,z1]) .| < e by A2,A3,Th3;
A5: 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 x1,x2,y1,y2,z1,z2 be Real;
     assume that
A6:  x1 in I & x2 in I & y1 in J & y2 in J & z1 in K & z2 in K and
A7:  |.x2-x1.| < r & |.y2-y1.| < r & |.z2-z1.| < r;

     [x1,y1] in [:I,J:] & [x2,y2] in [:I,J:] by A6,ZFMISC_1:87; then
     [x1,y1,z1] in E & [x2,y2,z2] in E by A6,ZFMISC_1:87;
     hence thesis by A4,A7;
    end;
    take r;
    thus thesis by A4,A5;
end;
