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 Th46:
  for f be PartFunc of the carrier of S, REAL st f
  is_Lipschitzian_on X holds f is_continuous_on X
proof
  let f be PartFunc of the carrier of S, REAL;
  assume
A1: f is_Lipschitzian_on X;
  then consider r such that
A2: 0<r and
A3: for x1,x2 st x1 in X & x2 in X holds |.f/.x1-f/.x2.|<=r*||. x1-x2
  .||;
A4: X c= dom f by A1;
  then
A5: dom (f|X) = X by RELAT_1:62;
  now
    let x0 such that
A6: x0 in X;
    now
      let g such that
A7:   0<g;
      reconsider s=g/r as Real;
      take s9=s;
A8:   now
        let x1;
        assume that
A9:     x1 in dom (f|X) and
A10:    ||. x1- x0 .|| <s;
        r*||. x1- x0 .|| <(g/r)*r by A2,A10,XREAL_1:68;
        then
A11:    r*||. x1- x0 .|| <g by A2,XCMPLX_1:87;
        |. f/.x1- f/.x0 .| <=r*||. x1- x0 .|| by A3,A5,A6,A9;
        then |. f/.x1- f/.x0 .| <g by A11,XXREAL_0:2;
        then |. (f|X)/.x1-f/.x0 .|<g by A9,PARTFUN2:15;
        hence |. (f|X)/.x1-(f|X)/.x0 .|<g by A5,A6,PARTFUN2:15;
      end;
      0<r" & s9=g*r" by A2,XCMPLX_0:def 9;
      hence
      0<s9 & for x1 st x1 in dom (f|X) & ||. x1- x0 .|| <s9 holds |. (f
      |X)/.x1-(f|X)/.x0.|<g by A7,A8,XREAL_1:129;
    end;
    hence f|X is_continuous_in x0 by A5,A6,Th8;
  end;
  hence thesis by A4;
end;
