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
  (for x0 st x0 in X holds f.x0 = |.x0.|) implies f|X is continuous
proof
  assume that
A1: for x0 st x0 in X holds f.x0 = |.x0.|;
  now
    let x0;
    assume
A2: x0 in dom(f|X);
    then f.x0=|.x0.| by A1;
    hence (f|X).x0=|.x0.| by A2,FUNCT_1:47;
  end;
  hence thesis by Th44;
end;
