
theorem Th18:
  for S being non empty non void TopStruct for f being
Collineation of S for X being Subset of S st X is closed_under_lines holds f.:X
  is closed_under_lines
proof
  let S be non empty non void TopStruct;
  let f be Collineation of S;
  let X be Subset of S;
  assume
A1: X is closed_under_lines;
  thus f.:X is closed_under_lines
  proof
    let l be Block of S;
    assume 2 c= card(l /\ (f.:X));
    then consider a,b being object such that
A2: a in l /\ (f.:X) and
A3: b in l /\ (f.:X) and
A4: a<>b by PENCIL_1:2;
    b in f.:X by A3,XBOOLE_0:def 4;
    then consider B being object such that
A5: B in dom f and
A6: B in X and
A7: b=f.B by FUNCT_1:def 6;
    b in l by A3,XBOOLE_0:def 4;
    then B in (f"l) by A5,A7,FUNCT_1:def 7;
    then
A8: B in (f"l) /\ X by A6,XBOOLE_0:def 4;
    a in f.:X by A2,XBOOLE_0:def 4;
    then consider A being object such that
A9: A in dom f and
A10: A in X and
A11: a=f.A by FUNCT_1:def 6;
    a in l by A2,XBOOLE_0:def 4;
    then A in (f"l) by A9,A11,FUNCT_1:def 7;
    then A in (f"l) /\ X by A10,XBOOLE_0:def 4;
    then 2 c= card ((f"l) /\ X) by A4,A11,A7,A8,PENCIL_1:2;
    then f"l c= X by A1;
    then
A12: f.:(f"l) c= f.:X by RELAT_1:123;
    f is bijective by Def4;
    then
A13: rng f = the carrier of S by FUNCT_2:def 3;
    l in the topology of S;
    hence thesis by A13,A12,FUNCT_1:77;
  end;
end;
