reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,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 Th14:
  X c= dom f implies (f|X is continuous iff for x0,r st x0 in X &
0<r ex s st 0<s & for x1 st x1 in X & |.x1-x0.| < s holds |.f.x1-f.x0.| < r
  )
proof
  assume
A1: X c= dom f;
  thus f|X is continuous implies for x0,r st x0 in X & 0<r ex s st 0<s & for
  x1 st x1 in X & |.x1-x0.| < s holds |.f.x1-f.x0.| < r
  proof
    assume
A2: f|X is continuous;
    let x0,r;
    assume that
A3: x0 in X and
A4: 0<r;
    x0 in dom(f|X) by A1,A3,RELAT_1:62;
    then f|X is_continuous_in x0 by A2;
    then consider s such that
A5: 0<s and
A6: for x1 st x1 in dom(f|X) & |.x1-x0.|<s holds |.(f|X).x1-(f|X).
    x0.| <r by A4,Th3;
    take s;
    thus 0<s by A5;
    let x1;
    assume that
A7: x1 in X and
A8: |.x1-x0.|<s;
A9: dom (f|X) = dom f /\ X by RELAT_1:61
      .= X by A1,XBOOLE_1:28;
    then |.f.x1-f.x0.| = |.(f|X).x1 - f.x0.| by A7,FUNCT_1:47
      .= |.(f|X).x1 - (f|X).x0.| by A3,A9,FUNCT_1:47;
    hence thesis by A6,A9,A7,A8;
  end;
  assume
A10: for x0,r st x0 in X & 0<r ex s st 0<s & for x1 st x1 in X & |.x1-
  x0.| < s holds |.f.x1-f.x0.| < r;
  now
    let x0 such that
A11: x0 in dom(f|X);
A12: x0 in X by A11;
    for r st 0<r ex s st 0<s & for x1 st x1 in dom(f|X) & |.x1-x0.|<s
    holds |.(f|X).x1-(f|X).x0.|<r
    proof
      let r;
      assume 0<r;
      then consider s such that
A13:  0<s and
A14:  for x1 st x1 in X & |.x1-x0.| < s holds |.f.x1-f.x0.| < r by A10,A12;
      take s;
      thus 0<s by A13;
      let x1 such that
A15:  x1 in dom(f|X) and
A16:  |.x1-x0.|<s;
      |.(f|X).x1-(f|X).x0.| = |.(f|X).x1 - f.x0.| by A11,FUNCT_1:47
        .= |.f.x1-f.x0.| by A15,FUNCT_1:47;
      hence thesis by A14,A15,A16;
    end;
    hence f|X is_continuous_in x0 by Th3;
  end;
  hence thesis;
end;
