
theorem
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_uniformly_continuous_on dom f & f = g holds
  ProjPMap1(g,t) is uniformly_continuous
& ProjPMap2(g,t) is uniformly_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_uniformly_continuous_on dom f and
A2:  f = g;

    for r be Real st 0 < r ex s be Real st 0 < s &
     for y1,y2 be Real st y1 in dom ProjPMap1(g,t)
       & y2 in dom ProjPMap1(g,t) & |. y1-y2 .| < s
      holds |. ProjPMap1(g,t).y1 - ProjPMap1(g,t).y2 .| < r
    proof
     let r be Real;
     assume 0 < r; then
     consider s be Real such that
A3:   0 < s and
A4:   for p1,p2 be Point of [:RNS_Real,RNS_Real:]
       st p1 in dom f & p2 in dom f & ||. p1-p2 .|| < s
        holds ||. f/.p1 - f/.p2 .|| < r by A1;

     now let y1,y2 be Real;
      assume that
A5:    y1 in dom ProjPMap1(g,t) and
A6:    y2 in dom ProjPMap1(g,t) and
A7:    |. y1-y2 .| < s;

A8:   X-section(dom g,t) = {y where y is Element of REAL : [t,y] in dom g}
        by MEASUR11:def 4;
A9:  y1 in X-section(dom g,t) & y2 in X-section(dom g,t)
        by A5,A6,MESFUN12:def 3; then
A10:  ex y be Element of REAL st y = y1 & [t,y] in dom g by A8;
A11:  ex y be Element of REAL st y = y2 & [t,y] in dom g by A8,A9;
      reconsider xx = t as Point of RNS_Real;
      reconsider yy1 = y1 as Point of RNS_Real by XREAL_0:def 1;
      reconsider yy2 = y2 as Point of RNS_Real by XREAL_0:def 1;
      reconsider p1 = [xx,yy1] as Point of [:RNS_Real,RNS_Real:];
      reconsider p2 = [xx,yy2] as Point of [:RNS_Real,RNS_Real:];

A12:  yy1-yy2 = y1-y2 by DUALSP03:4;
      p1-p2 = [xx,yy1] + [-xx,-yy2] by PRVECT_3:18; then
      p1-p2 = [xx-xx,yy1-yy2] by PRVECT_3:18; then
      p1-p2 = [0.RNS_Real,yy1-yy2] by RLVECT_1:15; then
      ||. p1-p2 .|| = sqrt( ||. 0.RNS_Real .||^2 + ||. yy1-yy2 .||^2 )
        by NDIFF_8:1; then
      ||. p1-p2 .|| = ||. yy1-yy2 .|| by SQUARE_1:22; then
      ||. p1-p2 .|| = |. y1-y2 .| by A12,EUCLID:def 2; then
A13:  ||. f/.p1 - f/.p2 .|| < r by A4,A7,A2,A10,A11;

      ProjPMap1(g,t).y1 = g.(t,y1) & ProjPMap1(g,t).y2 = g.(t,y2)
      by A10,A11,MESFUN12:def 3; then
      ProjPMap1(g,t).y1 = f/.p1 & ProjPMap1(g,t).y2 = f/.p2
        by A2,A10,A11,PARTFUN1:def 6; then
      ProjPMap1(g,t).y1 - ProjPMap1(g,t).y2 = f/.p1 - f/.p2
        by DUALSP03:4;
      hence |. ProjPMap1(g,t).y1 - ProjPMap1(g,t).y2 .| < r
        by A13,EUCLID:def 2;
     end;
     hence thesis by A3;
    end;
    hence ProjPMap1(g,t) is uniformly_continuous by FCONT_2:def 1;

    for r be Real st 0 < r ex s be Real st 0 < s &
     for x1,x2 be Real st x1 in dom ProjPMap2(g,t)
       & x2 in dom ProjPMap2(g,t) & |. x1-x2 .| < s
      holds |. ProjPMap2(g,t).x1 - ProjPMap2(g,t).x2 .| < r
    proof
     let r be Real;
     assume 0 < r; then
     consider s be Real such that
A14:   0 < s and
A15:   for p1,p2 be Point of [:RNS_Real,RNS_Real:]
       st p1 in dom f & p2 in dom f & ||. p1-p2 .|| < s
        holds ||. f/.p1 - f/.p2 .|| < r by A1;

     now let x1,x2 be Real;
      assume that
A16:    x1 in dom ProjPMap2(g,t) and
A17:    x2 in dom ProjPMap2(g,t) and
A18:    |. x1-x2 .| < s;

A19:   Y-section(dom g,t) = {x where x is Element of REAL : [x,t] in dom g}
        by MEASUR11:def 5;
A20:  x1 in Y-section(dom g,t) & x2 in Y-section(dom g,t)
        by A16,A17,MESFUN12:def 4; then
A21:  ex x be Element of REAL st x = x1 & [x,t] in dom g by A19;
A22:  ex x be Element of REAL st x = x2 & [x,t] in dom g by A19,A20;
      reconsider yy = t as Point of RNS_Real;
      reconsider xx1 = x1 as Point of RNS_Real by XREAL_0:def 1;
      reconsider xx2 = x2 as Point of RNS_Real by XREAL_0:def 1;
      reconsider p1 = [xx1,yy] as Point of [:RNS_Real,RNS_Real:];
      reconsider p2 = [xx2,yy] as Point of [:RNS_Real,RNS_Real:];

A23:  xx1-xx2 = x1-x2 by DUALSP03:4;
      p1-p2 = [xx1,yy] + [-xx2,-yy] by PRVECT_3:18; then
      p1-p2 = [xx1-xx2,yy-yy] by PRVECT_3:18; then
      p1-p2 = [xx1-xx2,0.RNS_Real] by RLVECT_1:15; then
      ||. p1-p2 .|| = sqrt( ||. 0.RNS_Real .||^2 + ||. xx1-xx2 .||^2 )
        by NDIFF_8:1; then
      ||. p1-p2 .|| = ||. xx1-xx2 .|| by SQUARE_1:22; then
      ||. p1-p2 .|| = |. x1-x2 .| by A23,EUCLID:def 2; then
A24:  ||. f/.p1 - f/.p2 .|| < r by A15,A18,A2,A21,A22;

      ProjPMap2(g,t).x1 = g.(x1,t) & ProjPMap2(g,t).x2 = g.(x2,t)
      by A21,A22,MESFUN12:def 4; then
      ProjPMap2(g,t).x1 = f/.p1 & ProjPMap2(g,t).x2 = f/.p2
        by A2,A21,A22,PARTFUN1:def 6; then
      ProjPMap2(g,t).x1 - ProjPMap2(g,t).x2 = f/.p1 - f/.p2
        by DUALSP03:4;
      hence |. ProjPMap2(g,t).x1 - ProjPMap2(g,t).x2 .| < r
        by A24,EUCLID:def 2;
     end;
     hence thesis by A14;
    end;
    hence ProjPMap2(g,t) is uniformly_continuous by FCONT_2:def 1;
end;
