
theorem
for f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
 p2 be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
 z be Element of REAL
st f is_uniformly_continuous_on dom f & f = g & p2 = ProjPMap2(g,z) holds
  p2 is_uniformly_continuous_on dom p2
proof
    let f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
    p2 be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    z be Element of REAL;
    assume that
A1: f is_uniformly_continuous_on dom f and
A2: f = g and
A3: p2 = ProjPMap2(g,z);

    reconsider zz=z as Point of RNS_Real;
A4: Y-section(dom g,z)
     = {t where t is Element of [:RNS_Real,RNS_Real:] : [t,z] in dom g}
        by MEASUR11:def 5;

    now let r be Real;
     assume 0 < r; then
     consider s be Real such that
A5:  0 < s and
A6:  for v1,v2 be Point of [:[:RNS_Real,RNS_Real:],RNS_Real:] st
      v1 in dom f & v2 in dom f & ||. v1-v2 .|| < s holds
       ||. f/.v1 - f/.v2 .|| < r by A1;
     take s;
     thus 0 < s by A5;
     hereby let xy1,xy2 be Point of [:RNS_Real,RNS_Real:];
      assume that
A7:   xy1 in dom p2 & xy2 in dom p2 and
A8:   ||. xy1-xy2 .|| < s;

A9:   xy1 in Y-section(dom g,z) & xy2 in Y-section(dom g,z)
        by A3,A7,MESFUN12:def 4; then
A10:   ex t1 be Element of [:RNS_Real,RNS_Real:] st t1 = xy1 & [t1,z] in dom g
        by A4;
A11:  ex t2 be Element of [:RNS_Real,RNS_Real:] st t2 = xy2 & [t2,z] in dom g
        by A4,A9;

      consider xx1 be Point of RNS_Real, yy1 be Point of RNS_Real such that
A12:  xy1 = [xx1,yy1] by PRVECT_3:18;
      consider xx2 be Point of RNS_Real, yy2 be Point of RNS_Real such that
A13:  xy2 = [xx2,yy2] by PRVECT_3:18;
      reconsider x1=xx1, y1=yy1, x2=xx2, y2=yy2 as Real;
      reconsider v1 = [xx1,yy1,zz], v2 = [xx2,yy2,zz]
       as Point of [:[:RNS_Real,RNS_Real:],RNS_Real:];

      v1-v2 = [xx1,yy1,zz] + [-xx2,-yy2,-zz] by PRVECT_4:9; then
      v1-v2 = [xx1-xx2,yy1-yy2,zz-zz] by PRVECT_4:9; then
      v1-v2 = [xx1-xx2,yy1-yy2,0.RNS_Real] by RLVECT_1:15; then
A14:  ||.v1-v2.||
         = sqrt(||.xx1-xx2.||^2 + ||.yy1-yy2.||^2 + ||. 0.RNS_Real .||^2)
           by PRVECT_4:9;

      xy1-xy2 = [xx1,yy1] + [-xx2,-yy2] by A12,A13,PRVECT_3:18; then
      xy1-xy2 = [xx1-xx2,yy1-yy2] by PRVECT_3:18; then
      ||.xy1-xy2.|| = sqrt(||.xx1-xx2.||^2 + ||.yy1-yy2.||^2)
         by NDIFF_8:1; then
A15:  ||. f/.v1 - f/.v2 .|| < r by A2,A6,A8,A10,A11,A12,A13,A14;

      p2/.xy1 = ProjPMap2(g,z).xy1 by A3,A7,PARTFUN1:def 6; then
      p2/.xy1 = g.([x1,y1],z) by A10,A12,MESFUN12:def 4; then
A16:  p2/.xy1 = f/.v1 by A2,A10,A12,PARTFUN1:def 6;
      p2/.xy2 = ProjPMap2(g,z).xy2 by A3,A7,PARTFUN1:def 6; then
      p2/.xy2 = g.([x2,y2],z) by A13,A11,MESFUN12:def 4;
      hence ||. p2/.xy1 - p2/.xy2 .|| < r by
        A2,A11,A13,A15,A16,PARTFUN1:def 6;
     end;
    end;
    hence p2 is_uniformly_continuous_on dom p2;
end;
