reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;

theorem Th74:
  proj1.:Cl P c= Cl(proj1.:P)
proof
  let y be object;
  assume y in proj1.:Cl P;
  then consider x being object such that
A1: x in the carrier of TOP-REAL 2 and
A2: x in Cl P and
A3: y = proj1.x by FUNCT_2:64;
  reconsider x as Point of TOP-REAL 2 by A1;
  set r = proj1.x;
  for O being open Subset of REAL st y in O holds O /\ proj1.:P is non empty
  proof
    reconsider e = x as Point of Euclid 2 by TOPREAL3:8;
    let O be open Subset of REAL;
    assume y in O;
    then consider g being Real such that
A4: 0 < g and
A5: ].r-g,r+g.[ c= O by A3,RCOMP_1:19;
    reconsider g as Real;
    reconsider B = Ball(e,g/2) as Subset of TOP-REAL 2 by TOPREAL3:8;
A6: B is open by GOBOARD6:3;
    e in B by A4,TBSP_1:11,XREAL_1:139;
    then P meets B by A2,A6,TOPS_1:12;
    then P /\ B <> {};
    then consider d being Point of TOP-REAL 2 such that
A7: d in P /\ B by SUBSET_1:4;
A8: d in B by A7,XBOOLE_0:def 4;
    then x`1-g/2 < d`1 by Th37;
    then
A9: r-g/2 < d`1 by PSCOMP_1:def 5;
    d`1 < x`1+g/2 by A8,Th37;
    then
A10: d`1 < r+g/2 by PSCOMP_1:def 5;
    d in P by A7,XBOOLE_0:def 4;
    then proj1.d in proj1.:P by FUNCT_2:35;
    then
A11: d`1 in proj1.:P by PSCOMP_1:def 5;
A12: g/2 < g/1 by A4,XREAL_1:76;
    then r-g < r-g/2 by XREAL_1:15;
    then
A13: r-g < d`1 by A9,XXREAL_0:2;
    r+g/2 < r+g by A12,XREAL_1:6;
    then
A14: d`1 < r+g by A10,XXREAL_0:2;
    ].r-g,r+g.[ = {t where t is Real: r-g < t & t < r+g}
            by RCOMP_1:def 2;
    then d`1 in ].r-g,r+g.[ by A13,A14;
    hence thesis by A5,A11,XBOOLE_0:def 4;
  end;
  hence thesis by A3,MEASURE6:63;
end;
