reserve

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

theorem Th10:
  for k being Nat for X being non empty set st 0 < k &
k + 1 c= card X for A being POINT of G_(k,X) for L being LINE of G_(k,X) holds
  A on L iff A c= L
proof
  let k be Nat;
  let X be non empty set;
  assume
A1: 0 < k & k + 1 c= card X;
  then
A2: the Points of G_(k,X) = {A where A is Subset of X: card A = k} by Def1;
  let A be POINT of G_(k,X);
  A in the Points of G_(k,X);
  then
A3: ex A1 being Subset of X st A1 = A & card A1 = k by A2;
A4: the Lines of G_(k,X) = {L where L is Subset of X: card L = k + 1} by A1
,Def1;
  let L be LINE of G_(k,X);
  L in the Lines of G_(k,X);
  then
A5: ex L1 being Subset of X st L1 = L & card L1 = k + 1 by A4;
A6: the Inc of G_(k,X) = (RelIncl bool X) /\ [:the Points of G_(k,X), the
  Lines of G_(k,X):] by A1,Def1;
  thus A on L implies A c= L
  proof
    assume A on L;
    then [A,L] in the Inc of G_(k,X);
    then [A,L] in RelIncl bool X by A6,XBOOLE_0:def 4;
    hence thesis by A3,A5,WELLORD2:def 1;
  end;
  thus A c= L implies A on L
  proof
    assume A c= L;
    then [A,L] in RelIncl bool X by A3,A5,WELLORD2:def 1;
    then [A,L] in the Inc of G_(k,X) by A6,XBOOLE_0:def 4;
    hence thesis;
  end;
end;
