theorem Th41:
  i => j in FI & j => k in FI implies i => k in FI
proof
  assume that
A1: i => j in FI and
A2: j => k in FI;
A3: FI \/ {i} c= <.FI \/ {i}.) by Def4;
  {i} c= FI \/ {i} by XBOOLE_1:7;
  then
A4: {i} c= <.FI \/ {i}.) by A3;
  FI c= FI \/ {i} by XBOOLE_1:7;
  then
A5: FI c= <.FI \/ {i}.) by A3;
  i in {i} by TARSKI:def 1;
  then j in <.FI \/ {i}.) by A1,A5,A4,Th29;
  then
A6: k in <.FI \/ {i}.) by A2,A5,Th29;
  <.FI.) = FI by Th21;
  hence thesis by A6,Th40;
end;
