reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

theorem Th33:
  f is Lipschitzian iff
  ex r be Real st 0<r & for x1,x2 st x1 in dom f &
  x2 in dom f holds |.f/.x1-f/.x2.| <= r*|.x1-x2.|
proof
  hereby assume f is Lipschitzian;
  then
  consider g be PartFunc of REAL,REAL-NS n such that
A1:  g=f & g is Lipschitzian;
  consider r be Real such that
A2: 0<r & for x1,x2 st x1 in dom g &
    x2 in dom g holds ||.g/.x1-g/.x2.||<=r*|.x1-x2.| by A1;
  take r;
  thus 0<r by A2;
  thus for x1,x2 st x1 in dom f &
       x2 in dom f holds |.f/.x1-f/.x2.|<=r*|.x1-x2.|
  proof
    let x1,x2;
    assume x1 in dom f & x2 in dom f; then
A3: ||.g/.x1-g/.x2.||<=r*|.x1-x2.| by A1,A2;
    f/.x1 = g/.x1 &  f/.x2 = g/.x2 by A1,REAL_NS1:def 4;
    hence |.f/.x1-f/.x2.|<=r*|.x1-x2.| by A3,REAL_NS1:1,5;
  end;
end;
  given r be Real such that
A4: 0<r & for x1,x2 st x1 in dom f &
    x2 in dom f holds |.f/.x1-f/.x2.|<=r*|.x1-x2.|;
  reconsider g= f as PartFunc of REAL,REAL-NS n
  by REAL_NS1:def 4;
  now
    let x1,x2;
    assume x1 in dom g & x2 in dom g; then
A5: |.f/.x1-f/.x2.|<=r*|.x1-x2.| by A4;
    f/.x1 = g/.x1 &  f/.x2 = g/.x2 by REAL_NS1:def 4;
    hence ||.g/.x1-g/.x2.||<=r*|.x1-x2.| by A5,REAL_NS1:1,5;
  end;
  then g is Lipschitzian by A4;
  hence thesis;
end;
