reserve

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

theorem Th32:
  for k being Nat for X being non empty set st 0 < k &
  k + 3 c= card X for F being IncProjMap over G_(k+1,X), G_(k+1,X) st F is
automorphism for H being IncProjMap over G_(k,X), G_(k,X) st the line-map of H
  = the point-map of F for f being Permutation of X st the IncProjMap of H =
  incprojmap(k,f) holds the IncProjMap of F = incprojmap(k+1,f)
proof
  let k be Nat;
  let X be non empty set such that
A1: 0 < k and
A2: k + 3 c= card X;
  k + 1 <= k + 3 by XREAL_1:7;
  then Segm(k + 1) c= Segm(k + 3) by NAT_1:39;
  then
A3: k + 1 c= card X by A2;
  then
A4: the Lines of G_(k,X) = {L where L is Subset of X: card L = k + 1} by A1
,Def1;
  k + 2 <= k + 3 by XREAL_1:7;
  then Segm(k + 2) c= Segm(k + 3) by NAT_1:39;
  then
A5: (k + 1) + 1 c= card X by A2;
  then
A6: the Points of G_(k+1,X) = {A where A is Subset of X: card A = k + 1} by
Def1;
  k + 0 <= k + 1 by XREAL_1:7;
  then
A7: Segm k c= Segm(k + 1) by NAT_1:39;
  k + 1 <= k + 2 by XREAL_1:7;
  then
A8: Segm(k + 1) c= Segm(k + 2) by NAT_1:39;
  let F be IncProjMap over G_(k+1,X), G_(k+1,X) such that
A9: F is automorphism;
A10: F is incidence_preserving by A9;
  let H be IncProjMap over G_(k,X), G_(k,X) such that
A11: the line-map of H = the point-map of F;
A12: dom(the point-map of F) = the Points of G_(k+1,X) by FUNCT_2:52;
  let f be Permutation of X such that
A13: the IncProjMap of H = incprojmap(k,f);
A14: 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+1,f)).x
  proof
    let x be object;
    assume x in dom(the point-map of F);
    then consider A being POINT of G_(k+1,X) such that
A15: x = A;
    consider A1 being LINE of G_(k,X) such that
A16: x = A1 by A4,A6,A15;
    incprojmap(k,f).A1 = f.:A1 by A1,A3,Def14;
    then F.A = incprojmap(k+1,f).A by A11,A13,A5,A15,A16,Def14;
    hence thesis by A15;
  end;
A17: the Lines of G_(k+1,X) = {L where L is Subset of X: card L = (k + 1) +
  1} by A5,Def1;
A18: the point-map of F is bijective by A9;
A19: 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+1,f)).x
  proof
    let x be object;
    assume x in dom(the line-map of F);
    then consider A being LINE of G_(k+1,X) such that
A20: x = A;
    reconsider x as set by TARSKI:1;
    x in the Lines of G_(k+1,X) by A20;
    then
A21: ex A11 being Subset of X st x = A11 & card A11 = (k + 1) + 1 by A17;
    then consider B1 being set such that
A22: B1 c= x and
A23: card B1 = k + 1 by A8,CARD_FIL:36;
A24: x is finite by A21;
    then
A25: card(x \ B1) = (k + 2) - (k + 1) by A21,A22,A23,CARD_2:44;
    B1 c= X by A21,A22,XBOOLE_1:1;
    then B1 in the Points of G_(k+1,X) by A6,A23;
    then consider b1 being POINT of G_(k+1,X) such that
A26: b1 = B1;
    consider C1 being set such that
A27: C1 c= B1 and
A28: card C1 = k by A7,A23,CARD_FIL:36;
    B1 is finite by A23;
    then
A29: card(C1 \/ (x \ B1)) = k + 1 by A27,A28,A24,A25,CARD_2:40,XBOOLE_1:85;
    C1 c= x by A22,A27;
    then
A30: C1 \/ (x \ B1) c= x by XBOOLE_1:8;
    then C1 \/ (x \ B1) c= X by A21,XBOOLE_1:1;
    then C1 \/ (x \ B1) in the Points of G_(k+1,X) by A6,A29;
    then consider b2 being POINT of G_(k+1,X) such that
A31: b2 = C1 \/ (x \ B1);
    b2 on A by A5,A20,A30,A31,Th10;
    then F.b2 on F.A by A10;
    then
A32: F.b2 c= F.A by A5,Th10;
    B1 \/ (C1 \/ (x \ B1)) c= x by A22,A30,XBOOLE_1:8;
    then
A33: card(b1 \/ b2) c= k + 2 by A21,A26,A31,CARD_1:11;
    B1 misses (x \ B1) by XBOOLE_1:79;
    then card((x \ B1) /\ B1) = 0 by CARD_1:27,XBOOLE_0:def 7;
    then
A34: b1 <> b2 by A25,A26,A31,XBOOLE_1:11,28;
    then (k + 1) + 1 c= card(b1 \/ b2) by A23,A29,A26,A31,Th1;
    then card(b1 \/ b2) = k + 2 by A33,XBOOLE_0:def 10;
    then
A35: b1 \/ b2 = x by A21,A22,A24,A30,A26,A31,CARD_2:102,XBOOLE_1:8;
    F.b2 in the Points of G_(k+1,X);
    then
A36: ex B12 being Subset of X st F.b2 = B12 & card B12 = k + 1 by A6;
    F.b1 in the Points of G_(k+1,X);
    then
A37: ex B11 being Subset of X st F.b1 = B11 & card B11 = k + 1 by A6;
    F.A in the Lines of G_(k+1,X);
    then
A38: ex L1 being Subset of X st F.A = L1 & card L1 = (k + 1) + 1 by A17;
    then
A39: F.A is finite;
    F.b1 <> F.b2 by A18,A12,A34,FUNCT_1:def 4;
    then
A40: (k + 1) + 1 c= card(F.b1 \/ F.b2) by A37,A36,Th1;
    b1 on A by A5,A20,A22,A26,Th10;
    then F.b1 on F.A by A10;
    then
A41: F.b1 c= F.A by A5,Th10;
    then F.b1 \/ F.b2 c= F.A by A32,XBOOLE_1:8;
    then card(F.b1 \/ F.b2) c= k + 2 by A38,CARD_1:11;
    then card(F.b1 \/ F.b2) = k + 2 by A40,XBOOLE_0:def 10;
    then
A42: F.b1 \/ F.b2 = F.A by A41,A32,A38,A39,CARD_2:102,XBOOLE_1:8;
A43: incprojmap(k+1,f).A = f.:x by A5,A20,Def14;
A44: f.:b1 \/ f.:b2 = f.:(b1 \/ b2) & F.b2 = incprojmap(k+1,f).b2 by A12,A14,
RELAT_1:120;
    F.b1 = incprojmap(k+1,f).b1 & incprojmap(k+1,f).b1 = f.:b1 by A5,A12,A14
,Def14;
    hence thesis by A5,A20,A35,A42,A43,A44,Def14;
  end;
A45: dom(the line-map of F) = the Lines of G_(k+1,X) & dom(the line-map of
  incprojmap(k+1,f)) = the Lines of G_(k+1,X) by FUNCT_2:52;
  dom(the point-map of incprojmap(k+1,f)) = the Points of G_(k+1,X) by
FUNCT_2:52;
  then the point-map of F = the point-map of incprojmap(k+1,f) by A12,A14;
  hence thesis by A45,A19,FUNCT_1:def 11;
end;
