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

theorem
  f1 is_Lipschitzian_on X & f2 is_Lipschitzian_on X1 implies f1+f2
  is_Lipschitzian_on X /\ X1
proof
  assume that
A1: f1 is_Lipschitzian_on X and
A2: f2 is_Lipschitzian_on X1;
A3: f1 is_Lipschitzian_on X /\ X1 by A1,Th38,XBOOLE_1:17;
  then consider s such that
A4: 0<s and
A5: for x1,x2 st x1 in X /\ X1 & x2 in X /\ X1 holds ||. f1/.x1-f1/.x2
  .||<=s*||. x1-x2 .||;
A6: f2 is_Lipschitzian_on X /\ X1 by A2,Th38,XBOOLE_1:17;
  then
A7: X /\ X1 c= dom f2;
  X /\ X1 c= dom f1 by A3;
  then X /\ X1 c= dom f1 /\ dom f2 by A7,XBOOLE_1:19;
  hence
A8: X /\ X1 c= dom (f1+f2) by VFUNCT_1:def 1;
  consider g such that
A9: 0<g and
A10: for x1,x2 st x1 in X /\ X1 & x2 in X /\ X1 holds ||. f2/.x1-f2/.x2
  .||<=g*||. x1-x2 .|| by A6;
  take p=s+g;
  0+0<s+g by A4,A9;
  hence 0<p;
  let x1,x2;
  assume that
A11: x1 in X/\X1 and
A12: x2 in X/\X1;
A13: ||. f2/.x1-f2/.x2.||<=g*||. x1-x2.|| by A10,A11,A12;
  ||. f1/.x1-f1/.x2.||<=s*||. x1-x2.|| by A5,A11,A12;
  then
A14: ||. f1/.x1-f1/.x2.||+||. f2/.x1-f2/.x2.||<= s*||. x1-x2.||+g*||. x1-x2
  .|| by A13,XREAL_1:7;
  ||. (f1+f2)/.x1-(f1+f2)/.x2 .|| = ||. f1/.x1 + f2/.x1-(f1+f2)/.x2.|| by A8
,A11,VFUNCT_1:def 1
    .= ||. f1/.x1 + f2/.x1 - (f1/.x2+f2/.x2) .|| by A8,A12,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;
  then
  ||. (f1+f2)/.x1-(f1+f2)/.x2.|| <=||. f1/.x1-f1/.x2.||+||. f2/.x1-f2/.x2
  .|| by NORMSP_1:def 1;
  hence thesis by A14,XXREAL_0:2;
end;
