
theorem
  for L1, L2, T1, T2 being antisymmetric reflexive non empty RelStr, f
  being Function of L1, T1, g being Function of L2, T2 st f is
  filtered-infs-preserving & g is filtered-infs-preserving holds [:f, g:] is
  filtered-infs-preserving
proof
  let L1, L2, T1, T2 be antisymmetric reflexive non empty RelStr, f be
  Function of L1, T1, g be Function of L2, T2 such that
A1: f is filtered-infs-preserving and
A2: g is filtered-infs-preserving;
  let X be Subset of [:L1, L2:];
  assume
A3: X is non empty filtered;
  then proj1 X is non empty filtered by YELLOW_3:21,24;
  then
A4: f preserves_inf_of proj1 X by A1;
  proj2 X is non empty filtered by A3,YELLOW_3:21,24;
  then
A5: g preserves_inf_of proj2 X by A2;
  set iX = [:f, g:].:X;
A6: dom f = the carrier of L1 & dom g = the carrier of L2 by FUNCT_2:def 1;
  assume
A7: ex_inf_of X, [:L1, L2:];
  then
A8: ex_inf_of proj1 X, L1 by YELLOW_3:42;
  X c= the carrier of [:L1, L2:];
  then
A9: X c= [:the carrier of L1, the carrier of L2:] by YELLOW_3:def 2;
  then
A10: proj2 iX = g.:proj2 X by A6,Th4;
A11: ex_inf_of proj2 X, L2 by A7,YELLOW_3:42;
  then
A12: ex_inf_of proj2 iX, T2 by A5,A10;
A13: proj1 iX = f.:proj1 X by A6,A9,Th4;
  then ex_inf_of proj1 iX, T1 by A4,A8;
  hence ex_inf_of ([:f, g:].:X), [:T1, T2:] by A12,YELLOW_3:42;
  hence inf ([:f, g:].:X) = [inf (f.:proj1 X), inf (g.:proj2 X)] by A13,A10,Th7
    .= [f.inf proj1 X, inf (g.:proj2 X)] by A4,A8
    .= [f.inf proj1 X, g.inf proj2 X] by A5,A11
    .= [:f, g:].(inf proj1 X, inf proj2 X) by A6,FUNCT_3:def 8
    .= [:f, g:].inf X by A7,Th7;
end;
