
theorem Th31:
for I,J,K be non empty closed_interval Subset of REAL, x,y be Element of REAL,
 f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL
 st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
 holds for e be Real st 0 < e holds
     ex r be Real st 0 < r
      & for u1,u2 be Element of [:REAL,REAL:], x1,y1,x2,y2 be Real
         st u1=[x1,y1] & u2=[x2,y2] & |.x2-x1.| < r & |.y2-y1.| < r
          & u1 in [:I,J:] & u2 in [:I,J:]
         holds for z be Element of REAL st z in K
         holds |. ProjPMap1(R_EAL g,u2).z-ProjPMap1(R_EAL g,u1).z .| < e
proof
    let I,J,K be non empty closed_interval Subset of REAL,
    x,y be Element 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: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g;

A4:dom(R_EAL g) = [:[:I,J:],K:] by A1,A3,MESFUNC5:def 7;

    let e be Real;
    assume 0 < e; then
    consider r be Real such that
A5: 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 by A2,A3,Th8;

    take r;
    thus 0 < r by A5;

    let u1,u2 be Element of [:REAL,REAL:], x1,y1,x2,y2 be Real;
    assume
A6:u1=[x1,y1] & u2=[x2,y2] & |.x2-x1.| < r & |.y2-y1.| < r
  & u1 in [:I,J:] & u2 in [:I,J:];

    let z be Element of REAL;
A7:x1 in I & x2 in I & y1 in J & y2 in J by A6,ZFMISC_1:87;
    assume
A8:z in K;

    |. z-z .| < r by A5; then
    |. g.([x2,y2,z])-g.([x1,y1,z]) .| < e by A5,A6,A7,A8; then
A9: |. g.([u2,z])-g.([u1,z]).| < e by A6;
a9: g.([u2,z])-g.([u1,z]) = g.([u2,z]) qua ExtReal -g.([u1,z]);

    (ProjPMap1(R_EAL g,u1)).z = (R_EAL g).(u1,z)
  & (R_EAL g).(u1,z) = g.([u1,z])
  & (ProjPMap1(R_EAL g,u2)).z = (R_EAL g).(u2,z)
  & (R_EAL g).(u2,z) = g.([u2,z])
      by A4,A6,A8,ZFMISC_1:87,MESFUNC5:def 7,MESFUN12:def 3;
    hence thesis by A9,a9,EXTREAL1:12;
end;
