reserve n,m for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,t,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th10:
  f|X is Lipschitzian implies f|X is uniformly_continuous
proof
  assume f|X is Lipschitzian;
  then consider r be Real such that
A1: 0<r and
A2: for x1,x2 be Real st x1 in dom(f|X) & x2 in dom(f|X) holds
  |.f.x1-f.x2.|<=r*|.x1-x2.| by FCONT_1:32;
    for r st 0<r
     ex s st 0<s &
     for x1,x2 st x1 in dom(f|X) & x2 in dom(f|X) & |.x1-x2.| < s
       holds |.f.x1-f.x2.| < r
    proof
::   now    dlaczego nie akceptuje Diffuse Statement !!! A.T.
    let p such that
A3: 0<p;
    take s=p/r;
    thus 0<s by A1,A3,XREAL_1:139;
    let x1,x2;
    assume that
A4: x1 in dom(f|X) and
A5: x2 in dom(f|X) and
A6: |.x1-x2.| < s;
A7: r*|.x1-x2.| < s*r by A1,A6,XREAL_1:68;
    |.f.x1-f.x2.|<=r*|.x1-x2.| by A2,A4,A5;
    then |.f.x1-f.x2.|<p/r*r by A7,XXREAL_0:2;
    hence |.f.x1-f.x2.| < p by A1,XCMPLX_1:87;
  end;
 hence thesis by Th1;
end;
