reserve i,j,k,n for Nat,
  X,Y,a,b,c,x for set,
  r,s for Real;
reserve f,g for FinSequence of TOP-REAL 2;

theorem Th11:
  for f,g st f,g are_in_general_position holds L~f /\ L~g is finite
proof
  let f,g such that
A1: f,g are_in_general_position;
  set BL = { LSeg(g,j) : 1 <= j & j+1 <= len g };
  set AL = { LSeg(f,i) : 1 <= i & i+1 <= len f };
A2: now
    let Z be set;
    assume Z in INTERSECTION(AL,BL);
    then consider X,Y be set such that
A3: X in AL & Y in BL and
A4: Z = X /\ Y by SETFAM_1:def 5;
    ( ex i be Nat st X = LSeg(f,i) & 1 <= i & i+1 <= len f)& ex
    j be Nat st Y = LSeg(g,j) & 1 <= j & j+1 <= len g by A3;
    hence Z is finite by A1,A4,Th9;
  end;
  L~f /\ L~g = union INTERSECTION(AL,BL) & INTERSECTION(AL,BL) is finite
  by Th10,SETFAM_1:28;
  hence thesis by A2,FINSET_1:7;
end;
