reserve Y for TopStruct;
reserve X for non empty TopSpace;
reserve X for almost_discrete non empty TopSpace;
reserve X,Y for non empty TopSpace;
reserve X for discrete non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve X for almost_discrete non empty TopSpace,
  X0 for maximal_discrete non empty SubSpace of X;

theorem Th70:
  for r being continuous Function of X,X0 holds r is
being_a_retraction implies for F being Subset of X0, E being Subset of X st F =
  E holds r" F = Cl E
proof
  let r be continuous Function of X,X0;
  reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
A1: A is maximal_discrete by Th45;
  assume
A2: r is being_a_retraction;
A3: for a being Point of X holds A /\ Cl {a} = {r.a}
  proof
    let a be Point of X;
A4: {a} c= Cl {a} by PRE_TOPC:18;
    consider c being Point of X such that
A5: c in A and
A6: A /\ Cl {a} = {c} by A1,Th58;
    reconsider b = c as Point of X0 by A5;
    {c} c= Cl {a} by A6,XBOOLE_1:17;
    then
A7: c in Cl {a} by ZFMISC_1:31;
    Cl {c} c= r" {b} by A2,Lm4;
    then Cl {a} c= r" {b} by A7,Th49;
    then {a} c= r" {b} by A4,XBOOLE_1:1;
    then a in r" {b} by ZFMISC_1:31;
    then r.a in {b} by FUNCT_2:38;
    hence thesis by A6,TARSKI:def 1;
  end;
  let F be Subset of X0, E be Subset of X;
  set R = {Cl {a} where a is Point of X : a in E};
  assume
A8: F = E;
  now
    let x be object;
    assume
A9: x in r" F;
    then reconsider b = x as Point of X;
A10: r.b in F by A9,FUNCT_2:38;
    then reconsider a = r.b as Point of X by A8;
    Cl {a} in R by A8,A10;
    then
A11: Cl {a} c= union R by ZFMISC_1:74;
A12: {b} c= Cl {b} by PRE_TOPC:18;
    A /\ Cl {b} = {a} by A3;
    then a in A /\ Cl {b} by ZFMISC_1:31;
    then a in Cl {b} by XBOOLE_0:def 4;
    then
A13: Cl {a} = Cl {b} by Th49;
    b in {b} by TARSKI:def 1;
    then b in Cl {a} by A13,A12;
    hence x in union R by A11;
  end;
  then
A14: r" F c= union R by TARSKI:def 3;
A15: A is discrete by A1;
  now
    let C be set;
    assume C in R;
    then consider a being Point of X such that
A16: C = Cl {a} and
A17: a in E;
    now
      let x be object;
      assume
A18:  x in C;
      then reconsider b = x as Point of X by A16;
A19:  A /\ Cl {b} = {r.b} by A3;
A20:  A /\ Cl {a} = {a} by A8,A15,A17,Th36;
      Cl {a} = Cl {b} by A16,A18,Th49;
      then a = r.x by A20,A19,ZFMISC_1:3;
      hence x in r" F by A8,A16,A17,A18,FUNCT_2:38;
    end;
    hence C c= r" F by TARSKI:def 3;
  end;
  then
A21: union R c= r" F by ZFMISC_1:76;
  Cl E = union R by Th48;
  hence thesis by A21,A14;
end;
