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 Th42:
  (for x0 st x0 in dom f holds f.x0 = x0^2) implies f|dom f is continuous
proof
  reconsider f1 = id dom f as PartFunc of REAL,REAL;
  assume
A1: for x0 st x0 in dom f holds f.x0 = x0^2;
A2: now
    let x0 be object;
    assume
A3: x0 in dom f;
    then reconsider x1 = x0 as Real;
    thus f.x0 = x1^2 by A1,A3
      .= f1.x1*x1 by A3,FUNCT_1:18
      .= f1.x0*f1.x0 by A3,FUNCT_1:18;
  end;
  dom f1 /\ dom f1 = dom f;
  then f = f1(#)f1 by A2,VALUED_1:def 4;
  hence thesis;
end;
