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

theorem Th26:
f|X is Lipschitzian
iff ex r be Real st 0<r & for x1,x2 st x1 in dom(f|X) & x2 in dom(f|X)
     holds ||. f/.x1-f/.x2 .|| <= r*|.x1-x2.|
proof
   thus f|X is Lipschitzian implies ex r be Real
         st 0<r & for x1,x2 st x1 in dom(f|X) &
            x2 in dom(f|X) holds ||.f/.x1-f/.x2.|| <= r*|.x1-x2.|
   proof
    given r be Real such that
A1:  0<r and
A2:  for x1,x2 st x1 in dom(f|X) & x2 in dom(f|X)
         holds ||.(f|X)/.x1-(f|X)/.x2.||<=r*|.x1-x2.|;
    take r;
    thus 0<r by A1;
    let x1,x2;
A3:   x1 in REAL & x2 in REAL by XREAL_0:def 1;
    assume A4: x1 in dom(f|X) & x2 in dom(f|X); then
    (f|X)/.x1 = f/.x1 & (f|X)/.x2 = f/.x2 by A3,PARTFUN2:15;
    hence thesis by A2,A4;
   end;
   given r be Real such that
A5: 0<r and
A6: for x1,x2 st x1 in dom(f|X) & x2 in dom(f|X)
        holds ||.f/.x1-f/.x2.|| <= r*|.x1-x2.|;
   take r;
   thus 0<r by A5;
   let x1,x2;
A7:   x1 in REAL & x2 in REAL by XREAL_0:def 1;
   assume A8: x1 in dom(f|X) & x2 in dom(f|X); then
   (f|X)/.x1 = f/.x1 & (f|X)/.x2 = f/.x2 by A7,PARTFUN2:15;
   hence thesis by A6,A8;
end;
