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
  X c= dom f & (for x0 st x0 in X holds f.x0 = x0^2) implies f|X is continuous
proof
  assume that
A1: X c= dom f and
A2: for x0 st x0 in X holds f.x0=x0^2;
  X=dom f /\ X by A1,XBOOLE_1:28;
  then
A3: X=dom(f|X) by RELAT_1:61;
  now
    let x0;
    assume
A4: x0 in dom(f|X);
    then f.x0=x0^2 by A2;
    hence (f|X).x0=x0^2 by A4,FUNCT_1:47;
  end;
  then f|X|X is continuous by A3,Th42;
  hence thesis;
end;
