reserve

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

theorem Th25:
  for k being Nat for X being non empty set st 1 < k &
  card X = k + 1 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
  let k be Nat;
  let X be non empty set such that
A1: 1 < k and
A2: k + 1 = card X;
  deffunc CH(object) = X \ {$1};
  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: X is finite by A2;
A6: the Points of G_(k,X) = {A where A is Subset of X: card A = k} by A1,A2
,Def1;
A7: 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
A8: x in dom c and
A9: y = c.x by FUNCT_1:def 3;
A10: card{x} = 1 by CARD_1:30;
    {x} c= X by A3,A8,ZFMISC_1:31;
    then
A11: card(X \ {x}) = (k + 1) - 1 by A2,A5,A10,CARD_2:44;
    y = X \ {x} by A3,A4,A8,A9;
    hence thesis by A6,A11;
  end;
  let F be IncProjMap over G_(k,X), G_(k,X);
  assume F is automorphism;
  then
A12: the point-map of F is bijective;
  reconsider c as Function of X, the Points of G_(k,X) by A3,A7,FUNCT_2:2;
  deffunc W(Element of X) = union (X \ F.(c.$1));
  consider f being Function such that
A13: dom f = X and
A14: 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
A15: a in X and
A16: b = f.a by A13,FUNCT_1:def 3;
    reconsider a as Element of X by A15;
A17: b = union (X \ F.(c.a)) by A14,A16;
    consider A being POINT of G_(k,X) such that
A18: A = F.(c.a);
    A in the Points of G_(k,X);
    then ex A1 being Subset of X st A1 = A & card A1 = k by A6;
    then card(X \ A) = (k + 1) - k by A2,A5,CARD_2:44;
    then consider x being object such that
A19: X \ A = {x} by CARD_2:42;
    x in X by A19,ZFMISC_1:31;
    hence thesis by A17,A18,A19,ZFMISC_1:25;
  end;
  then reconsider f as Function of X,X by A13,FUNCT_2:2;
A20: dom(the point-map of F) = the Points of G_(k,X) by FUNCT_2:52;
A21: f is one-to-one
  proof
    let x1,x2 be object such that
A22: x1 in dom f & x2 in dom f and
A23: f.x1 = f.x2;
    reconsider x1,x2 as Element of X by A22;
    consider A1 being POINT of G_(k,X) such that
A24: A1 = F.(c.x1);
    consider A2 being POINT of G_(k,X) such that
A25: A2 = F.(c.x2);
    A2 in the Points of G_(k,X);
    then
A26: ex A12 being Subset of X st A12 = A2 & card A12 = k by A6;
    then card(X \ A2) = (k + 1) - k by A2,A5,CARD_2:44;
    then consider y2 being object such that
A27: X \ A2 = {y2} by CARD_2:42;
    A1 in the Points of G_(k,X);
    then
A28: ex A11 being Subset of X st A11 = A1 & card A11 = k by A6;
    then card(X \ A1) = (k + 1) - k by A2,A5,CARD_2:44;
    then consider y1 being object such that
A29: X \ A1 = {y1} by CARD_2:42;
    f.x2 = union(X \ F.(c.x2)) by A14;
    then
A30: f.x2 = y2 by A25,A27,ZFMISC_1:25;
    f.x1 = union(X \ F.(c.x1)) by A14;
    then f.x1 = y1 by A24,A29,ZFMISC_1:25;
    then A1 = A2 by A23,A28,A26,A29,A27,A30,Th5;
    then
A31: c.x1 = c.x2 by A12,A20,A24,A25,FUNCT_1:def 4;
    c.x1 = X \ {x1} & c.x2 = X \ {x2} by A4;
    then {x1} = {x2} by A31,Th5;
    hence thesis by ZFMISC_1:3;
  end;
A32: rng(the point-map of F) = the Points of G_(k,X) by A12,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
A33: {y} c= X by ZFMISC_1:31;
    card{y} = 1 by CARD_1:30;
    then card(X \ {y}) = (k + 1) - 1 by A2,A5,A33,CARD_2:44;
    then X \ {y} in rng(the point-map of F) by A6,A32;
    then consider a being object such that
A34: a in dom(the point-map of F) and
A35: X \ {y} = (the point-map of F).a by FUNCT_1:def 3;
    reconsider a as set by TARSKI:1;
    a in the Points of G_(k,X) by A34;
    then
A36: ex A1 being Subset of X st A1 = a & card A1 = k by A6;
    then card(X \ a) = (k + 1) - k by A2,A5,CARD_2:44;
    then consider x being object such that
A37: X \ a = {x} by CARD_2:42;
    reconsider x as Element of X by A37,ZFMISC_1:31;
    a /\ X = X \ {x} by A37,XBOOLE_1:48;
    then
A38: X \ {x} = a by A36,XBOOLE_1:28;
    c.x = X \ {x} by A4;
    then X /\ {y} = X \ F.(c.x) by A35,A38,XBOOLE_1:48;
    then {y} = X \ F.(c.x) by A33,XBOOLE_1:28;
    then y = union(X \ F.(c.x)) by ZFMISC_1:25;
    then y = f.x by A14;
    hence thesis;
  end;
  then
A39: rng f = X by FUNCT_2:10;
  then f is onto by FUNCT_2:def 3;
  then reconsider f as Permutation of X by A21;
A40: dom(the line-map of F) = the Lines of G_(k,X) by FUNCT_2:52;
A41: 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
A42: x in dom(the point-map of F);
    then consider A being POINT of G_(k,X) such that
A43: x = A;
    F.A in the Points of G_(k,X);
    then
A44: ex B being Subset of X st B = F.A & card B = k by A6;
    then card(X \ F.A) = (k + 1) - k by A2,A5,CARD_2:44;
    then
A45: ex x2 being object st X \ F.A = {x2} by CARD_2:42;
    X \ (X \ F.A) = F.A /\ X & F.A /\ X = F.A by A44,XBOOLE_1:28,48;
    then
A46: F.A = X \ {union(X \ F.A)} by A45,ZFMISC_1:25;
A47: f.:X = X by A39,RELSET_1:22;
A48: ex A1 being Subset of X st x = A1 & card A1 = k by A6,A20,A42;
    then
A49: X \ (X \ A) = A /\ X & A /\ X = A by A43,XBOOLE_1:28,48;
    card(X \ A) = (k + 1) - k by A2,A5,A43,A48,CARD_2:44;
    then consider x1 being object such that
A50: X \ A = {x1} by CARD_2:42;
    reconsider x1 as Element of X by A50,ZFMISC_1:31;
A51: c.x1 = X \ {x1} & Im(f,x1) = {f.x1} by A4,A13,FUNCT_1:59;
    incprojmap(k,f).A = f.:A by A1,A2,Def14;
    then incprojmap(k,f).A = f.:X \ f.:{x1} by A50,A49,FUNCT_1:64;
    hence thesis by A14,A43,A50,A46,A49,A51,A47;
  end;
  dom(the point-map of incprojmap(k,f)) = the Points of G_(k,X) by FUNCT_2:52;
  then
A52: the point-map of F = the point-map of incprojmap(k,f) by A20,A41;
A53: the Lines of G_(k,X) = {L where L is Subset of X: card L = k + 1} by A1,A2
,Def1;
A54: 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
A55: x in dom(the line-map of F);
    then consider A being LINE of G_(k,X) such that
A56: x = A;
    F.A in the Lines of G_(k,X);
    then ex y being Subset of X st y = F.A & card y = k + 1 by A53;
    then
A57: F.A = X by A2,A5,CARD_2:102;
    ex A11 being Subset of X st x = A11 & card A11 = k + 1 by A53,A40,A55;
    then
A58: x = X by A2,A5,CARD_2:102;
    reconsider xx=x as set by TARSKI:1;
    incprojmap(k,f).A = f.:xx by A1,A2,A56,Def14;
    hence thesis by A39,A56,A58,A57,RELSET_1:22;
  end;
  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 A40,A52,A54,FUNCT_1:def 11;
  hence thesis;
end;
