reserve n,m for Nat,
  x,X,X1 for set,
  s,g,r,p for Real,
  S,T for RealNormSpace,
  f,f1,f2 for PartFunc of S, T,
  s1,s2,q1 for sequence of S,
  x0,x1, x2 for Point of S,
  Y for Subset of S;

theorem Th4:
  f is_uniformly_continuous_on X implies p(#)f is_uniformly_continuous_on X
proof
  assume
A1: f is_uniformly_continuous_on X;
  then X c= dom f;
  hence
A2: X c= dom (p(#)f) by VFUNCT_1:def 4;
  now
    per cases;
    suppose
A3:   p=0;
      let r;
      assume
A4:   0<r;
      then consider s such that
A5:   0<s and
      for x1,x2 st x1 in X & x2 in X & ||.x1-x2.||<s holds ||.f/.x1-f/.x2
      .||<r by A1;
      take s;
      thus 0<s by A5;
      let x1,x2;
      assume that
A6:   x1 in X and
A7:   x2 in X and
      ||.x1-x2.||<s;
      ||.(p(#)f)/.x1-(p(#)f)/.x2.|| = ||.p*(f/.x1)-(p(#)f)/.x2.|| by A2,A6,
VFUNCT_1:def 4
        .= ||.0.T -(p(#)f)/.x2.|| by A3,RLVECT_1:10
        .= ||.0.T - p*(f/.x2).|| by A2,A7,VFUNCT_1:def 4
        .= ||.0.T - 0.T.|| by A3,RLVECT_1:10
        .= ||.0.T .|| by RLVECT_1:13
        .= 0 by NORMSP_0:def 6;
      hence ||.(p(#)f)/.x1-(p(#)f)/.x2.|| <r by A4;
    end;
    suppose
A8:   p<>0;
      then
A9:   0 <> |.p.| by COMPLEX1:47;
      let r;
A10:  0<|.p.| by A8,COMPLEX1:47;
      assume 0<r;
      then 0 < r/|.p.| by A10,XREAL_1:139;
      then consider s such that
A11:  0<s and
A12:  for x1,x2 st x1 in X & x2 in X & ||.x1-x2.||<s holds ||.f/.x1-f
      /.x2 .||<r/|.p.| by A1;
      take s;
      thus 0<s by A11;
      let x1,x2;
      assume that
A13:  x1 in X and
A14:  x2 in X and
A15:  ||.x1-x2.||<s;
A16:  ||.(p(#)f)/.x1-(p(#)f)/.x2.|| = ||.p*(f/.x1)-(p(#)f)/.x2.|| by A2,A13,
VFUNCT_1:def 4
        .= ||.p*(f/.x1) - p*(f/.x2).|| by A2,A14,VFUNCT_1:def 4
        .= ||.p*(f/.x1 - f/.x2).|| by RLVECT_1:34
        .= |.p.|*||.f/.x1 - f/.x2.|| by NORMSP_1:def 1;
      |.p.|*||.f/.x1-f/.x2.||<r/|.p.|*|.p.| by A10,A12,A13,A14,A15,
XREAL_1:68;
      hence ||.(p(#)f)/.x1-(p(#)f)/.x2.|| <r by A9,A16,XCMPLX_1:87;
    end;
  end;
  hence thesis;
end;
