reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;

theorem
  SFX is_finer_than SFY & SFY is_finer_than SFZ implies SFX is_finer_than SFZ
proof
  assume that
A1: for X st X in SFX ex Y st Y in SFY & X c= Y and
A2: for X st X in SFY ex Y st Y in SFZ & X c= Y;
  let X;
  assume X in SFX;
  then consider Y such that
A3: Y in SFY and
A4: X c= Y by A1;
  consider Z such that
A5: Z in SFZ & Y c= Z by A2,A3;
  take Z;
  thus thesis by A4,A5;
end;
