reserve

  k,n for Nat,
  x,y,X,Y,Z for set;

theorem Th24:
  for k being Nat for X being non empty set st k = 1 &
  k + 1 c= card X for F being IncProjMap over G_(k,X), G_(k,X) st F is
  automorphism holds ex s being Permutation of X st the IncProjMap of F =
  incprojmap(k,s)
proof
  deffunc CH(object) = {$1};
  let k be Nat;
  let X be non empty set such that
A1: k = 1 & k + 1 c= card X;
A2: the Points of G_(k,X) = {A where A is Subset of X: card A = 1} by A1,Def1;
  consider c being Function such that
A3: dom c = X and
A4: for x being object st x in X holds c.x = CH(x) from FUNCT_1:sch 3;
A5: rng c c= the Points of G_(k,X)
  proof
    let y be object;
    assume y in rng c;
    then consider x being object such that
A6: x in dom c & y = c.x by FUNCT_1:def 3;
A7: card {x} = 1 by CARD_1:30;
    {x} c= X & y = {x} by A3,A4,A6,ZFMISC_1:31;
    hence thesis by A2,A7;
  end;
  let F be IncProjMap over G_(k,X), G_(k,X) such that
A8: F is automorphism;
A9: the point-map of F is bijective by A8;
  reconsider c as Function of X, the Points of G_(k,X) by A3,A5,FUNCT_2:2;
  deffunc W(Element of X) = union (F.(c.$1));
  consider f being Function such that
A10: dom f = X and
A11: for x being Element of X holds f.x = W(x) from FUNCT_1:sch 4;
  rng f c= X
  proof
    let b be object;
    assume b in rng f;
    then consider a being object such that
A12: a in X and
A13: b = f.a by A10,FUNCT_1:def 3;
    reconsider a as Element of X by A12;
A14: b = union (F.(c.a)) by A11,A13;
    consider A being POINT of G_(k,X) such that
A15: A = F.(c.a);
    A in the Points of G_(k,X);
    then
A16: ex A1 being Subset of X st A1 = A & card A1 = 1 by A2;
    then consider x being object such that
A17: A = {x} by CARD_2:42;
    x in X by A16,A17,ZFMISC_1:31;
    hence thesis by A14,A15,A17,ZFMISC_1:25;
  end;
  then reconsider f as Function of X,X by A10,FUNCT_2:2;
A18: F is incidence_preserving by A8;
A19: dom(the point-map of F) = the Points of G_(k,X) by FUNCT_2:52;
A20: f is one-to-one
  proof
    let x1,x2 be object such that
A21: x1 in dom f & x2 in dom f and
A22: f.x1 = f.x2;
    reconsider x1,x2 as Element of X by A21;
    consider A1 being POINT of G_(k,X) such that
A23: A1 = F.(c.x1);
    A1 in the Points of G_(k,X);
    then ex A11 being Subset of X st A11 = A1 & card A11 = 1 by A2;
    then consider y1 being object such that
A24: A1 = {y1} by CARD_2:42;
A25: c.x1 = {x1} & c.x2 = {x2} by A4;
    consider A2 being POINT of G_(k,X) such that
A26: A2 = F.(c.x2);
    A2 in the Points of G_(k,X);
    then ex A12 being Subset of X st A12 = A2 & card A12 = 1 by A2;
    then consider y2 being object such that
A27: A2 = {y2} by CARD_2:42;
    f.x2 = union(F.(c.x2)) by A11;
    then
A28: f.x2 = y2 by A26,A27,ZFMISC_1:25;
    f.x1 = union(F.(c.x1)) by A11;
    then f.x1 = y1 by A23,A24,ZFMISC_1:25;
    then c.x1 = c.x2 by A9,A19,A22,A23,A26,A24,A27,A28,FUNCT_1:def 4;
    hence thesis by A25,ZFMISC_1:3;
  end;
A29: rng(the point-map of F) = the Points of G_(k,X) by A9,FUNCT_2:def 3;
  for y being object st y in X ex x being object st x in X & y = f.x
  proof
    let y be object;
    assume y in X;
    then
A30: {y} c= X by ZFMISC_1:31;
    card{y} = 1 by CARD_1:30;
    then {y} in rng(the point-map of F) by A2,A29,A30;
    then consider a being object such that
A31: a in dom(the point-map of F) and
A32: {y} = (the point-map of F).a by FUNCT_1:def 3;
    a in the Points of G_(k,X) by A31;
    then
A33: ex A1 being Subset of X st A1 = a & card A1 = 1 by A2;
    then consider x being object such that
A34: a = {x} by CARD_2:42;
    reconsider x as Element of X by A33,A34,ZFMISC_1:31;
    {y} = F.(c.x) by A4,A32,A34;
    then y = union(F.(c.x)) by ZFMISC_1:25;
    then y = f.x by A11;
    hence thesis;
  end;
  then rng f = X by FUNCT_2:10;
  then f is onto by FUNCT_2:def 3;
  then reconsider f as Permutation of X by A20;
A35: dom(the line-map of F) = the Lines of G_(k,X) by FUNCT_2:52;
A36: the Lines of G_(k,X) = {L where L is Subset of X: card L = 1 + 1} by A1
,Def1;
A37: for x being object st x in dom(the line-map of F)
   holds (the line-map of F).x = (the line-map of incprojmap(k,f)).x
  proof
    let x be object;
    assume
A38: x in dom(the line-map of F);
    then consider A being LINE of G_(k,X) such that
A39: x = A;
    consider A11 being Subset of X such that
A40: x = A11 and
A41: card A11 = 2 by A36,A35,A38;
    consider x1,x2 being object such that
A42: x1 <> x2 and
A43: x = {x1,x2} by A40,A41,CARD_2:60;
    reconsider x1,x2 as Element of X by A40,A43,ZFMISC_1:32;
    c.x1 = {x1} by A4;
    then consider A1 being POINT of G_(k,X) such that
A44: A1 = {x1};
    c.x2 = {x2} by A4;
    then consider A2 being POINT of G_(k,X) such that
A45: A2 = {x2};
    A1 <> A2 by A42,A44,A45,ZFMISC_1:18;
    then
A46: F.A1 <> F.A2 by A9,A19,FUNCT_1:def 4;
    F.A2 in the Points of G_(k,X);
    then
A47: ex B2 being Subset of X st B2 = F.A2 & card B2 = 1 by A2;
    then consider y2 being object such that
A48: F.A2 = {y2} by CARD_2:42;
    A1 c= A by A39,A43,A44,ZFMISC_1:36;
    then A1 on A by A1,Th10;
    then F.A1 on F.A by A18;
    then
A49: F.A1 c= F.A by A1,Th10;
A50: incprojmap(k,f).A = f.:A & f.:(A1 \/ A2) = f.:A1 \/ f.:A2 by A1,Def14,
RELAT_1:120;
A51: A1 \/ A2 = A by A39,A43,A44,A45,ENUMSET1:1;
    F.A1 in the Points of G_(k,X);
    then
A52: ex B1 being Subset of X st B1 = F.A1 & card B1 = 1 by A2;
    then
A53: ex y1 being object st F.A1 = {y1} by CARD_2:42;
    A2 c= A by A39,A43,A45,ZFMISC_1:36;
    then A2 on A by A1,Th10;
    then F.A2 on F. A by A18;
    then
A54: F.A2 c= F.A by A1,Th10;
    F.(c.x2) = F.A2 by A4,A45;
    then
A55: f.x2 = union(F.A2) by A11;
    Im(f,x2) = {f.x2} by A10,FUNCT_1:59;
    then
A56: f.:A2 = F.A2 by A45,A55,A48,ZFMISC_1:25;
A57: F.A1 is finite by A52;
    not y2 in F.A1 by A46,A52,A47,A57,A48,CARD_2:102,ZFMISC_1:31;
    then
A58: card(F.A1 \/ F.A2) = 1 + 1 by A52,A53,A48,CARD_2:41;
    F.(c.x1) = F.A1 by A4,A44;
    then
A59: f.x1 = union(F.A1) by A11;
    Im(f,x1) = {f.x1} by A10,FUNCT_1:59;
    then
A60: f.:A1 = F .A1 by A44,A59,A53,ZFMISC_1:25;
    F.A in the Lines of G_(k,X);
    then
A61: ex B3 being Subset of X st B3 = F.A & card B3 = 2 by A36;
    then F.A is finite;
    hence thesis by A39,A50,A51,A49,A54,A61,A58,A60,A56,CARD_2:102,XBOOLE_1:8;
  end;
A62: for x being object st x in dom(the point-map of F)
    holds (the point-map of F).x = (the point-map of incprojmap(k,f)).x
  proof
    let x be object;
    assume
A63: x in dom(the point-map of F);
    then consider A being POINT of G_(k,X) such that
A64: x = A;
A65: ex A1 being Subset of X st x = A1 & card A1 = 1 by A2,A19,A63;
    then consider x1 being object such that
A66: x = {x1} by CARD_2:42;
    reconsider x1 as Element of X by A65,A66,ZFMISC_1:31;
    F.(c.x1) = F.A by A4,A64,A66;
    then
A67: f.x1 = union(F.A) by A11;
    F.A in the Points of G_(k,X);
    then consider B being Subset of X such that
A68: B = F.A and
A69: card B = 1 by A2;
A70: ex x2 being object st B = {x2} by A69,CARD_2:42;
    incprojmap(k,f).A = f.:A & Im(f,x1) = {f.x1} by A1,A10,Def14,FUNCT_1:59;
    hence thesis by A64,A66,A67,A68,A70,ZFMISC_1:25;
  end;
  dom(the point-map of incprojmap(k,f)) = the Points of G_(k,X) by FUNCT_2:52;
  then
A71: the point-map of F = the point-map of incprojmap(k,f) by A19,A62;
  dom(the line-map of incprojmap(k,f)) = the Lines of G_(k,X) by FUNCT_2:52;
  then the IncProjMap of F = incprojmap(k,f) by A35,A71,A37,FUNCT_1:def 11;
  hence thesis;
end;
