reserve p, q for Point of TOP-REAL 2,
  r for Real,
  h for non constant standard special_circular_sequence,
  g for FinSequence of TOP-REAL 2,
  f for non empty FinSequence of TOP-REAL 2,
  I, i1, i, j, k for Nat;

theorem Th17:
  for X being Subset of REAL st X = { q`1 : q in L~g } holds X = (
  proj1 | L~g).:the carrier of (TOP-REAL 2)|L~g
proof
  set T = (TOP-REAL 2)|L~g;
  set F = proj1 | L~g;
  let X be Subset of REAL such that
A1: X = { q`1 : q in L~g };
  thus X c= (proj1 | L~g).:the carrier of T
  proof
    let x be object;
    assume x in X;
    then consider q1 being Point of TOP-REAL 2 such that
A2: q1`1 = x and
A3: q1 in L~g by A1;
A4: x = F.q1 by A2,A3,PSCOMP_1:22;
A5: dom F = the carrier of T by FUNCT_2:def 1
      .= [#] ((TOP-REAL 2)|L~g)
      .= L~g by PRE_TOPC:def 5;
    then q1 in the carrier of T by A3,FUNCT_2:def 1;
    hence thesis by A3,A5,A4,FUNCT_1:def 6;
  end;
  thus (proj1 | L~g).:the carrier of T c= X
  proof
    let x be object;
    assume x in (proj1 | L~g).:the carrier of T;
    then consider x1 be object such that
    x1 in dom F and
A6: x1 in the carrier of T and
A7: x = F.x1 by FUNCT_1:def 6;
    x1 in [#] ((TOP-REAL 2)|L~g) by A6;
    then
A8: x1 in L~g by PRE_TOPC:def 5;
    then reconsider x2 = x1 as Element of TOP-REAL 2;
    x = x2`1 by A7,A8,PSCOMP_1:22;
    hence thesis by A1,A8;
  end;
end;
