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 Th1:
  x0 in X & f is_continuous_in x0 implies f|X is_continuous_in x0
proof
  assume that
A1: x0 in X and
A2: f is_continuous_in x0;
  let s1 such that
A3: rng s1 c= dom(f|X) and
A4: s1 is convergent & lim s1 = x0;
  dom(f|X) = X /\ dom f by RELAT_1:61;
  then
A5: rng s1 c= dom f by A3,XBOOLE_1:18;
A6: (f|X)/*s1 = f/*s1 by A3,FUNCT_2:117;
  hence (f|X)/*s1 is convergent by A2,A4,A5;
  thus (f|X).x0 = f.x0 by A1,FUNCT_1:49
    .= lim ((f|X)/*s1) by A2,A4,A5,A6;
end;
