reserve

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

theorem
  for X being non empty set st 0 < k & k + 1 c= card X for F being
  IncProjMap over G_(k,X), G_(k,X) holds F is automorphism iff ex s being
  Permutation of X st the IncProjMap of F = incprojmap(k,s)
proof
  let X be non empty set such that
A1: 0 < k and
A2: k + 1 c= card X;
  let F be IncProjMap over G_(k,X), G_(k,X);
A3: F is automorphism implies ex s being Permutation of X st the IncProjMap
  of F = incprojmap(k,s)
  proof
A4: card k = k & succ 1 = 1 + 1;
A5: card(k + 1) = k + 1;
    k + 1 in succ card X by A2,ORDINAL1:22;
    then
A6: k + 1 = card X or k + 1 in card X by ORDINAL1:8;
A7: card 1 = 1;
    0 + 1 < k + 1 & succ Segm k = Segm(k + 1) by A1,NAT_1:38,XREAL_1:8;
    then Segm 1 in succ Segm k by A7,A5,NAT_1:41;
    then 1 = k or Segm 1 in Segm k by ORDINAL1:8;
    then
A8: 1 = k or 1 < k & Segm 2 c= Segm k by A7,A4,NAT_1:41,ORDINAL1:21;
    assume
A9: F is automorphism;
    succ Segm(k + 1) = Segm(k + 1 + 1) by NAT_1:38;
    then 1 = k or 1 < k & card X = k + 1 or 2 <= k &
     k + 2 c= card X by A6,A8,
NAT_1:39,ORDINAL1:21;
    hence thesis by A2,A9,Th24,Th25,Th33;
  end;
  (ex s being Permutation of X st the IncProjMap of F = incprojmap(k,s))
  implies F is automorphism
  proof
    assume
    ex s being Permutation of X st the IncProjMap of F = incprojmap(k ,s);
    then consider s being Permutation of X such that
A10: the IncProjMap of F = incprojmap(k,s);
A11: incprojmap(k,s) is automorphism by A1,A2,Th34;
    then
A12: incprojmap(k,s) is incidence_preserving;
A13: F is incidence_preserving
    proof
      let A be POINT of G_(k,X);
      let L be LINE of G_(k,X);
      F.A = incprojmap(k,s).A & F.L = incprojmap(k,s).L by A10;
      hence thesis by A12;
    end;
    the line-map of F is bijective & the point-map of F is bijective by A10,A11
;
    hence thesis by A13;
  end;
  hence thesis by A3;
end;
