reserve

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

theorem Th34:
  for k being Nat for X being non empty set st 0 < k &
  k + 1 c= card X for s being Permutation of X holds incprojmap(k,s) is
  automorphism
proof
  let k be Nat;
  let X be non empty set such that
A1: 0 < k & k + 1 c= card X;
  let s be Permutation of X;
A2: the Points of G_(k,X) = {A where A is Subset of X: card A = k} by A1,Def1;
A3: the point-map of incprojmap(k,s) is one-to-one
  proof
    let x1,x2 be object;
    assume that
A4: x1 in dom (the point-map of incprojmap(k,s)) and
A5: x2 in dom (the point-map of incprojmap(k,s)) and
A6: (the point-map of incprojmap(k,s)).x1 = (the point-map of
    incprojmap(k,s)).x2;
    consider X1 being POINT of G_(k,X) such that
A7: X1 = x1 by A4;
    x1 in the Points of G_(k,X) by A4;
    then
A8: ex X11 being Subset of X st X11 = x1 & card X11 = k by A2;
    consider X2 being POINT of G_(k,X) such that
A9: X2 = x2 by A5;
    x2 in the Points of G_(k,X) by A5;
    then
A10: ex X12 being Subset of X st X12 = x2 & card X12 = k by A2;
A11: incprojmap(k,s).X2 = s.:X2 by A1,Def14;
    incprojmap(k,s).X1 = s.:X1 by A1,Def14;
    hence thesis by A6,A7,A9,A8,A10,A11,Th6;
  end;
  for y being object st y in the Points of G_(k,X)
ex x being object st x in the Points of G_(k,X )
  & y = (the point-map of incprojmap(k,s)).x
  proof
    let y be object;
    assume y in the Points of G_(k,X);
    then
A12: ex B being Subset of X st B = y & card B = k by A2;
     reconsider y as set by TARSKI:1;
A13: s"y c= dom s by RELAT_1:132;
    then
A14: s"y c= X by FUNCT_2:52;
    rng s = X by FUNCT_2:def 3;
    then
A15: s.:(s"y) = y by A12,FUNCT_1:77;
    then card(s"y) = k by A12,A13,Th4;
    then s"y in the Points of G_(k,X) by A2,A14;
    then consider A being POINT of G_(k,X) such that
A16: A = s"y;
    y = incprojmap(k,s).A by A1,A15,A16,Def14;
    hence thesis;
  end;
  then rng (the point-map of incprojmap(k,s)) = the Points of G_(k,X) by
FUNCT_2:10;
  then
A17: the point-map of incprojmap(k,s) is onto by FUNCT_2:def 3;
A18: the Lines of G_(k,X) = {L where L is Subset of X: card L = k + 1} by A1
,Def1;
  for y being object st y in the Lines of G_(k,X)
ex x being object st x in the Lines of G_(k,X) &
  y = (the line-map of incprojmap(k,s)).x
  proof
    let y be object;
    assume y in the Lines of G_(k,X);
    then
A19: ex B being Subset of X st B = y & card B = k + 1 by A18;
    reconsider y as set by TARSKI:1;
A20: s"y c= dom s by RELAT_1:132;
    then
A21: s"y c= X by FUNCT_2:52;
    rng s = X by FUNCT_2:def 3;
    then
A22: s.:(s"y) = y by A19,FUNCT_1:77;
    then card(s"y) = k + 1 by A19,A20,Th4;
    then s"y in the Lines of G_(k,X) by A18,A21;
    then consider A being LINE of G_(k,X) such that
A23: A = s"y;
    y = incprojmap(k,s).A by A1,A22,A23,Def14;
    hence thesis;
  end;
  then rng (the line-map of incprojmap(k,s)) = the Lines of G_(k,X) by
FUNCT_2:10;
  then
A24: the line-map of incprojmap(k,s) is onto by FUNCT_2:def 3;
A25: dom s = X by FUNCT_2:52;
A26: incprojmap(k,s) is incidence_preserving
  proof
    let A1 be POINT of G_(k,X);
    let L1 be LINE of G_(k,X);
A27: s.:A1 = incprojmap(k,s).A1 & s.:L1 = incprojmap(k,s).L1 by A1,Def14;
    A1 in the Points of G_(k,X);
    then
A28: ex a1 being Subset of X st A1 = a1 & card a1 = k by A2;
A29: incprojmap(k,s).A1 on incprojmap(k,s).L1 implies A1 on L1
    proof
      assume incprojmap(k,s).A1 on incprojmap(k,s).L1;
      then s.:A1 c= s.:L1 by A1,A27,Th10;
      then A1 c= L1 by A25,A28,FUNCT_1:87;
      hence thesis by A1,Th10;
    end;
    A1 on L1 implies incprojmap(k,s).A1 on incprojmap(k,s).L1
    proof
      assume A1 on L1;
      then A1 c= L1 by A1,Th10;
      then s.:A1 c= s.:L1 by RELAT_1:123;
      hence thesis by A1,A27,Th10;
    end;
    hence thesis by A29;
  end;
  the line-map of incprojmap(k,s) is one-to-one
  proof
    let x1,x2 be object;
    assume that
A30: x1 in dom (the line-map of incprojmap(k,s)) and
A31: x2 in dom (the line-map of incprojmap(k,s)) and
A32: (the line-map of incprojmap(k,s)).x1 = (the line-map of incprojmap
    (k,s)).x2;
    consider X1 being LINE of G_(k,X) such that
A33: X1 = x1 by A30;
    x1 in the Lines of G_(k,X) by A30;
    then
A34: ex X11 being Subset of X st X11 = x1 & card X11 = k + 1 by A18;
    consider X2 being LINE of G_(k,X) such that
A35: X2 = x2 by A31;
    x2 in the Lines of G_(k,X) by A31;
    then
A36: ex X12 being Subset of X st X12 = x2 & card X12 = k + 1 by A18;
A37: incprojmap(k,s).X2 = s.:X2 by A1,Def14;
    incprojmap(k,s).X1 = s.:X1 by A1,Def14;
    hence thesis by A32,A33,A35,A34,A36,A37,Th6;
  end;
  hence thesis by A24,A3,A17,A26;
end;
