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 Th23:
 for g be PartFunc of REAL,REAL-NS n,
     f be PartFunc of REAL,REAL n st g=f holds
  g is continuous iff f is continuous
proof
  let g be PartFunc of REAL,REAL-NS n,
      f be PartFunc of REAL,REAL n;
  assume A1: g=f;
  hereby assume g is continuous;
    then for x0 st x0 in dom f holds f is_continuous_in x0 by A1;
    hence f is continuous;
  end;
  assume
A2: f is continuous;
  now
    let x0;
    assume x0 in dom g;
    then f is_continuous_in x0 by A2,A1;
    hence g is_continuous_in x0 by A1;
  end;
  hence thesis;
end;
