reserve X,X1 for set,
  r,s for Real,
  z for Complex,
  RNS for RealNormSpace,
  CNS, CNS1,CNS2 for ComplexNormSpace;

theorem
  for f1,f2 be PartFunc of CNS,RNS st f1 is_uniformly_continuous_on X &
  f2 is_uniformly_continuous_on X1 holds f1+f2 is_uniformly_continuous_on X/\X1
proof
  let f1,f2 be PartFunc of CNS,RNS;
  assume that
A1: f1 is_uniformly_continuous_on X and
A2: f2 is_uniformly_continuous_on X1;
A3: f2 is_uniformly_continuous_on X /\ X1 by A2,Th2,XBOOLE_1:17;
  then
A4: X /\ X1 c= dom f2;
A5: f1 is_uniformly_continuous_on X /\ X1 by A1,Th2,XBOOLE_1:17;
  then X /\ X1 c= dom f1;
  then X /\ X1 c= dom f1 /\ dom f2 by A4,XBOOLE_1:19;
  hence
A6: X /\ X1 c= dom (f1+f2) by VFUNCT_1:def 1;
  let r;
  assume 0 < r;
  then
A7: 0<r/2;
  then consider s such that
A8: 0<s and
A9: for x1,x2 be Point of CNS st x1 in X /\ X1 & x2 in X /\X1 & ||.x1-
  x2.||<s holds ||.f1/.x1-f1/.x2.||<r/2 by A5;
  consider g be Real such that
A10: 0<g and
A11: for x1,x2 be Point of CNS st x1 in X /\ X1 & x2 in X /\ X1 & ||.x1-
  x2.|| < g holds ||.f2/.x1-f2/.x2.|| < r/2 by A3,A7;
  take p=min(s,g);
  thus 0<p by A8,A10,XXREAL_0:15;
  let x1,x2 be Point of CNS;
  assume that
A12: x1 in X/\X1 and
A13: x2 in X/\X1 and
A14: ||.x1-x2.||<p;
  p <= g by XXREAL_0:17;
  then ||.x1-x2.||<g by A14,XXREAL_0:2;
  then
A15: ||.f2/.x1-f2/.x2.||<r/2 by A11,A12,A13;
  p <= s by XXREAL_0:17;
  then ||.x1-x2.||<s by A14,XXREAL_0:2;
  then ||.f1/.x1-f1/.x2.||<r/2 by A9,A12,A13;
  then
A16: ||.f1/.x1-f1/.x2.||+||.f2/.x1-f2/.x2.||<r/2+r/2 by A15,XREAL_1:8;
A17: ||.f1/.x1 - f1/.x2 + (f2/.x1-f2/.x2).|| <= ||.f1/.x1-f1/.x2.|| + ||.f2
  /.x1-f2/.x2.|| by NORMSP_1:def 1;
  ||.(f1+f2)/.x1-(f1+f2)/.x2.|| = ||.f1/.x1 + f2/.x1-(f1+f2)/.x2.|| by A6,A12,
VFUNCT_1:def 1
    .= ||.f1/.x1 + f2/.x1 - (f1/.x2+f2/.x2).|| by A6,A13,VFUNCT_1:def 1
    .= ||.f1/.x1 + (f2/.x1 - (f1/.x2+f2/.x2)).|| by RLVECT_1:def 3
    .= ||.f1/.x1 + (f2/.x1 - f1/.x2-f2/.x2).|| by RLVECT_1:27
    .= ||.f1/.x1 + (-f1/.x2 + f2/.x1-f2/.x2).||
    .= ||.f1/.x1 + (-f1/.x2 + (f2/.x1-f2/.x2)).|| by RLVECT_1:def 3
    .= ||.f1/.x1 - f1/.x2 + (f2/.x1-f2/.x2).|| by RLVECT_1:def 3;
  hence thesis by A16,A17,XXREAL_0:2;
end;
