
theorem Th7:
  for T being non empty TopSpace-like TopRelStr holds T is lower
  iff {(uparrow F)` where F is Subset of T: F is finite} is Basis of T
proof
  let T be non empty TopSpace-like TopRelStr;
  set BB = the set of all (uparrow x)` where x is Element of T;
  BB c= bool the carrier of T
  proof
    let x be object;
    assume x in BB;
    then ex y being Element of T st x = (uparrow y)`;
    hence thesis;
  end;
  then reconsider BB as Subset-Family of T;
  reconsider T9 = T as non empty RelStr;
  set A = {(uparrow F)` where F is Subset of T: F is finite};
  reconsider BB as Subset-Family of T;
A1: A = FinMeetCl BB
  proof
    deffunc F(Element of T9) = uparrow $1;
    defpred P[object,object] means
     ex x being Element of T st x = $2 & $1 = (uparrow x)`;
    hereby
      deffunc F(Element of T9) = uparrow $1;
      let a be object;
      assume a in A;
      then consider F being Subset of T such that
A2:   a = (uparrow F)` and
A3:   F is finite;
      set Y = {uparrow x where x is Element of T: x in F};
      Y c= bool the carrier of T
      proof
        let a be object;
        assume a in Y;
        then ex x being Element of T st a = uparrow x & x in F;
        hence thesis;
      end;
      then reconsider Y as Subset-Family of T;
      reconsider Y as Subset-Family of T;
      uparrow F = union Y by YELLOW_9:4;
      then
A4:   a = Intersect COMPLEMENT Y by A2,YELLOW_8:6;
      reconsider Y9 = Y as Subset-Family of T9;
A5:   Y9 = {F(x) where x is Element of T9: x in F};
A6:   COMPLEMENT Y9 = {F(x)` where x is Element of T9: x in F} from
      YELLOW_9:sch 2(A5);
A7:   COMPLEMENT Y c= BB
      proof
        let b be object;
        assume b in COMPLEMENT Y;
        then ex x being Element of T st b = (uparrow x)` & x in F by A6;
        hence thesis;
      end;
      deffunc F(Element of T) = (uparrow $1)`;
      {F(x) where x is Element of T: x in F} is finite from FRAENKEL:sch
      21(A3);
      hence a in FinMeetCl BB by A7,A6,A4,CANTOR_1:def 3;
    end;
    let a be object;
    assume a in FinMeetCl BB;
    then consider Y being Subset-Family of T such that
A8: Y c= BB and
A9: Y is finite and
A10: a = Intersect Y by CANTOR_1:def 3;
A11: now
      let y be object;
      assume y in Y;
      then y in BB by A8;
      then ex x being Element of T st y = (uparrow x)`;
      hence ex b being object st b in the carrier of T & P[y,b];
    end;
    consider f being Function such that
A12: dom f = Y & rng f c= the carrier of T and
A13: for y being object st y in Y holds P[y,f.y] from FUNCT_1:sch 6(A11);
    reconsider F = rng f as Subset of T by A12;
A14: F is finite by A9,A12,FINSET_1:8;
    set X = {uparrow x where x is Element of T: x in F};
    X c= bool the carrier of T
    proof
      let a be object;
      assume a in X;
      then ex x being Element of T st a = uparrow x & x in F;
      hence thesis;
    end;
    then reconsider X as Subset-Family of T;
    reconsider X as Subset-Family of T;
    reconsider X9 = X as Subset-Family of T9;
A15: X9 = {F(x) where x is Element of T9: x in F};
A16: COMPLEMENT X9 = {F(x)` where x is Element of T9: x in F} from
    YELLOW_9:sch 2(A15);
A17: COMPLEMENT X = Y
    proof
      hereby
        let a be object;
        assume a in COMPLEMENT X;
        then consider x being Element of T9 such that
A18:    a = (uparrow x)` and
A19:    x in F by A16;
        consider y being object such that
A20:    y in Y and
A21:    x = f.y by A12,A19,FUNCT_1:def 3;
        ex z being Element of T st z = f.y & y = (uparrow z)` by A13,A20;
        hence a in Y by A18,A20,A21;
      end;
      let a be object;
      assume
A22:  a in Y;
      then consider z being Element of T such that
A23:  z = f.a and
A24:  a = (uparrow z)` by A13;
      z in F by A12,A22,A23,FUNCT_1:def 3;
      hence thesis by A16,A24;
    end;
    uparrow F = union X by YELLOW_9:4;
    then a = (uparrow F)` by A10,A17,YELLOW_8:6;
    hence thesis by A14;
  end;
  thus thesis by A1,YELLOW_9:23;
end;
