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 Th24:
  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,PARTFUN2:15
      .= |.(f|X)/.x1 - (f|X)/.x0 .| by A3,A9,PARTFUN2:15;
    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;
A11: dom (f|X) = dom f /\ X by RELAT_1:61
    .= X by A1,XBOOLE_1:28;
  now
    let x0 such that
A12: x0 in dom(f|X);
A13: x0 in X by A12,RELAT_1:57;
    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
A14:  0<s and
A15:  for x1 st x1 in X & |.x1-x0.| < s
        holds |. f/.x1 - f/.x0 .| < r by A10,A13;
      take s;
      thus 0<s by A14;
      let x1 such that
A16:  x1 in dom(f|X) and
A17:  |.x1-x0.|<s;
      |.(f|X)/.x1-(f|X)/.x0 .| = |.(f|X)/.x1 - f/.x0 .| by A12,PARTFUN2:15
        .= |. f/.x1 - f/.x0 .| by A16,PARTFUN2:15;
      hence thesis by A11,A15,A16,A17;
    end;
    hence f|X is_continuous_in x0 by Th3,A12;
  end;
  hence thesis;
end;
