
theorem Th33:
for f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
 g be PartFunc of [:REAL,REAL:],REAL, t be Element of REAL
st f is_continuous_on dom f & f = g holds
  ProjPMap1(g,t) is continuous & ProjPMap2(g,t) is continuous
proof
    let f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL, t be Element of REAL;
    assume that
A1:  f is_continuous_on dom f and
A2:  f = g;

    for y0 be Real st y0 in dom ProjPMap1(g,t) holds
     ProjPMap1(g,t) is_continuous_in y0
    proof
     let y0 be Real;
     assume y0 in dom ProjPMap1(g,t); then
A3:  y0 in X-section(dom g,t) by MESFUN12:def 3;

A4:  X-section(dom g,t) = {y where y is Element of REAL : [t,y] in dom g}
        by MEASUR11:def 4; then
A5:  ex y be Element of REAL st y = y0 & [t,y] in dom g by A3;
     reconsider xx = t as Point of RNS_Real;
     reconsider yy0 = y0 as Point of RNS_Real by XREAL_0:def 1;
     reconsider p0 = [xx,yy0] as Point of [:RNS_Real,RNS_Real:];

     for r be Real st 0 < r ex s be Real st 0 < s
      & for y1 be Real st y1 in dom ProjPMap1(g,t) & |. y1-y0 .| < s
          holds |. ProjPMap1(g,t).y1 - ProjPMap1(g,t).y0 .| < r
     proof
      let r be Real;
      assume 0 < r; then
      consider s be Real such that
A6:    0 < s and
A7:   for p1 be Point of [:RNS_Real,RNS_Real:] st p1 in dom f &
        ||. p1-p0 .|| < s holds ||. f/.p1 - f/.p0 .|| < r
          by A1,A2,A5,NFCONT_1:19;

      now let y1 be Real;
       assume that
A8:     y1 in dom ProjPMap1(g,t) and
A9:     |. y1-y0 .| < s;

       y1 in X-section(dom g,t) by A8,MESFUN12:def 3; then
A10:   ex y be Element of REAL st y = y1 & [t,y] in dom g by A4;
       reconsider yy1 = y1 as Point of RNS_Real by XREAL_0:def 1;
       reconsider p1 = [xx,yy1] as Point of [:RNS_Real,RNS_Real:];

A11:   yy1-yy0 = y1-y0 by DUALSP03:4;
       p1-p0 = [xx,yy1] + [-xx,-yy0] by PRVECT_3:18; then
       p1-p0 = [xx-xx,yy1-yy0] by PRVECT_3:18; then
       p1-p0 = [0.RNS_Real,yy1-yy0] by RLVECT_1:15; then
       ||. p1-p0 .|| = sqrt(||. 0.RNS_Real .||^2 + ||. yy1-yy0 .||^2)
         by NDIFF_8:1; then
       ||. p1-p0 .|| = ||. yy1-yy0 .|| by SQUARE_1:22; then
       ||. p1-p0 .|| = |. y1-y0 .| by A11,EUCLID:def 2; then
A12:   ||. f/.p1-f/.p0 .|| < r by A9,A2,A10,A7;

       ProjPMap1(g,t).y1 = g.(t,y1) & ProjPMap1(g,t).y0 = g.(t,y0)
         by A5,A10,MESFUN12:def 3; then
       ProjPMap1(g,t).y1 = f/.p1 & ProjPMap1(g,t).y0 = f/.p0
         by A2,A5,A10,PARTFUN1:def 6; then
       ProjPMap1(g,t).y1 - ProjPMap1(g,t).y0 = f/.p1 - f/.p0
         by DUALSP03:4;
       hence |. ProjPMap1(g,t).y1 - ProjPMap1(g,t).y0 .| < r
         by A12,EUCLID:def 2;
      end;
      hence thesis by A6;
     end;
     hence ProjPMap1(g,t) is_continuous_in y0 by FCONT_1:3;
    end;
    hence ProjPMap1(g,t) is continuous;

    for x0 be Real st x0 in dom ProjPMap2(g,t) holds
     ProjPMap2(g,t) is_continuous_in x0
    proof
     let x0 be Real;
     assume x0 in dom ProjPMap2(g,t); then
A13:  x0 in Y-section(dom g,t) by MESFUN12:def 4;

A14:  Y-section(dom g,t) = {x where x is Element of REAL : [x,t] in dom g}
        by MEASUR11:def 5; then
A15:  ex x be Element of REAL st x = x0 & [x,t] in dom g by A13;
     reconsider yy = t as Point of RNS_Real;
     reconsider xx0 = x0 as Point of RNS_Real by XREAL_0:def 1;
     reconsider p0 = [xx0,yy] as Point of [:RNS_Real,RNS_Real:];

     for r be Real st 0 < r ex s be Real st 0 < s
      & for x1 be Real st x1 in dom ProjPMap2(g,t) & |. x1-x0 .| < s
          holds |. ProjPMap2(g,t).x1 - ProjPMap2(g,t).x0 .| < r
     proof
      let r be Real;
      assume 0 < r; then
      consider s be Real such that
A16:    0 < s and
A17:   for p1 be Point of [:RNS_Real,RNS_Real:] st p1 in dom f &
        ||. p1-p0 .|| < s holds ||. f/.p1 - f/.p0 .|| < r
          by A1,A2,A15,NFCONT_1:19;

      now let x1 be Real;
       assume that
A18:     x1 in dom ProjPMap2(g,t) and
A19:     |. x1-x0 .| < s;

       x1 in Y-section(dom g,t) by A18,MESFUN12:def 4; then
A20:   ex x be Element of REAL st x = x1 & [x,t] in dom g by A14;
       reconsider xx1 = x1 as Point of RNS_Real by XREAL_0:def 1;
       reconsider p1 = [xx1,yy] as Point of [:RNS_Real,RNS_Real:];

A21:   xx1-xx0 = x1-x0 by DUALSP03:4;
       p1-p0 = [xx1,yy] + [-xx0,-yy] by PRVECT_3:18; then
       p1-p0 = [xx1-xx0,yy-yy] by PRVECT_3:18; then
       p1-p0 = [xx1-xx0,0.RNS_Real] by RLVECT_1:15; then
       ||. p1-p0 .|| = sqrt(||. 0.RNS_Real .||^2 + ||. xx1-xx0 .||^2)
         by NDIFF_8:1; then
       ||. p1-p0 .|| = ||. xx1-xx0 .|| by SQUARE_1:22; then
       ||. p1-p0 .|| = |. x1-x0 .| by A21,EUCLID:def 2; then
A22:   ||. f/.p1-f/.p0 .|| < r by A19,A2,A20,A17;

       ProjPMap2(g,t).x1 = g.(x1,t) & ProjPMap2(g,t).x0 = g.(x0,t)
         by A15,A20,MESFUN12:def 4; then
       ProjPMap2(g,t).x1 = f/.p1 & ProjPMap2(g,t).x0 = f/.p0
         by A2,A15,A20,PARTFUN1:def 6; then
       ProjPMap2(g,t).x1 - ProjPMap2(g,t).x0 = f/.p1 - f/.p0
         by DUALSP03:4;
       hence |. ProjPMap2(g,t).x1 - ProjPMap2(g,t).x0 .| < r
         by A22,EUCLID:def 2;
      end;
      hence thesis by A16;
     end;
     hence ProjPMap2(g,t) is_continuous_in x0 by FCONT_1:3;
    end;
    hence ProjPMap2(g,t) is continuous;
end;
