reserve

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

theorem Th28:
  for k being Element of NAT for X being non empty set st 0 < k &
k + 1 c= card X for T1,T2 being Subset of the Points of G_(k,X) st T1 is STAR &
  T2 is STAR & meet T1 = meet T2 holds T1 = T2
proof
  let k be Element of NAT;
  let X be non empty set such that
A1: 0 < k & k + 1 c= card X;
  let T1,T2 be Subset of the Points of G_(k,X) such that
A2: T1 is STAR and
A3: T2 is STAR and
A4: meet T1 = meet T2;
  consider S2 being Subset of X such that
A5: card S2 = k - 1 & T2 = {A where A is Subset of X: card A = k & S2 c=
  A} by A3;
A6: S2 = meet T2 by A1,A5,Th26;
  consider S1 being Subset of X such that
A7: card S1 = k - 1 & T1 = {A where A is Subset of X: card A = k & S1 c=
  A} by A2;
  S1 = meet T1 by A1,A7,Th26;
  hence thesis by A4,A7,A5,A6;
end;
