
theorem Th5:
  for X,Y be RealNormSpace, f be LinearOperator of X,Y holds
  f is_continuous_on the carrier of X
    iff
  f is_continuous_in 0.X
proof
  let X,Y be RealNormSpace, f be LinearOperator of X,Y;
A1:  f| (the carrier of X) = f;
A2:  dom f = the carrier of X by FUNCT_2:def 1;
  now assume A3:f is_continuous_in 0.X;
    for x be Point of X st x in the carrier of X
       holds f| (the carrier of X) is_continuous_in x by A3,Th4;
    hence f is_continuous_on the carrier of X by A2,NFCONT_1:def 7;
  end;
  hence thesis by A1,NFCONT_1:def 7;
end;
