
theorem
  for S being TopStruct for X,Y being Subset of S st X,Y are_joinable ex
  f being one-to-one FinSequence of bool the carrier of S st (X = f.1 & Y = f.(
  len f) & (for W being Subset of S st W in rng f holds W is closed_under_lines
strong) & for i being Element of NAT st 1 <= i & i < len f holds 2 c= card((f.i
  ) /\ (f.(i+1))))
proof
  defpred P[set,set] means 2 c= card($1 /\ $2);
  let S be TopStruct;
  let X,Y be Subset of S;
  assume X,Y are_joinable;
  then consider f being FinSequence of bool the carrier of S such that
A1: X = f.1 & Y = f.(len f) and
A2: for W being Subset of S st W in rng f holds W is closed_under_lines
  strong and
A3: for i being Element of NAT st 1 <= i & i < len f holds 2 c= card((f.
  i) /\ (f.(i+1)));
A4: for i being Element of NAT, d1,d2 being set st 1 <= i & i < len f & d1 =
  f.i & d2 = f.(i+1) holds P[d1,d2] by A3;
  consider g being one-to-one FinSequence of bool the carrier of S such that
A5: X = g.1 & Y = g.(len g) and
A6: rng g c= rng f and
A7: for i being Element of NAT st 1 <= i & i < len g holds P[g.i,g.(i+1)
  ] from FinSeqOneToOne(A1,A4);
  take g;
  thus thesis by A2,A5,A6,A7;
end;
