theorem Th15:
  (i/\/FI) "\/" (j/\/FI) = (i"\/"j)/\/FI & (i/\/FI) "/\" (j/\/FI)
  = (i"/\"j)/\/FI
proof
  set R = equivalence_wrt FI;
A1: j/\/FI = Class(R,j) by Def6;
  reconsider jj = join(I), mm = meet(I) as BinOp of R by Th13,Th14;
A2: i/\/FI = Class(R,i) by Def6;
A3: I /\/ FI = LattStr (#Class R, jj/\/R, mm/\/R#) by Def5;
  (i"\/"j)/\/FI = Class(R,i"\/"j) by Def6;
  hence (i/\/FI) "\/" (j/\/FI) = (i"\/"j)/\/FI by A2,A1,A3,Th3;
  (i"/\"j)/\/FI = Class(R,i"/\"j) by Def6;
  hence thesis by A2,A1,A3,Th3;
end;
