 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th10:
  for x being Point of T holds
    (the carrier of S) --> x is_continuous_on the carrier of S
proof
  let x be Point of T;
A1: now
    let x1 be Point of S;
    let r be Real;
    assume that x1 in the carrier of S and
A2: 0 < r;
    take s = r;
    thus 0 < s by A2;
    let x2 be Point of S;
    assume x2 in the carrier of S & ||.(x2 - x1).|| < s;
    reconsider xx1 = x1, xx2 = x2 as Point of S;
    ( ((the carrier of S) --> x) /. xx1 = x
      & ((the carrier of S) --> x) /. xx2 = x ) by FUNCOP_1:7;
    hence ||.((((the carrier of S) --> x) /. x2)
     - (((the carrier of S) --> x) /. x1)).|| < r by A2,NORMSP_1:6;
  end;
  dom ((the carrier of S) --> x) = the carrier of S by FUNCOP_1:13;
  hence (the carrier of S) --> x is_continuous_on the carrier of S
    by A1, NFCONT_1:19;
end;
