
theorem Th38:
  for L being up-complete Semilattice st SupMap L is
meet-preserving holds for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2)
  = sup (I1 "/\" I2)
proof
  let L be up-complete Semilattice such that
A1: SupMap L is meet-preserving;
  set f = SupMap L;
  let I1, I2 be Ideal of L;
  reconsider x = I1, y = I2 as Element of InclPoset(Ids L) by YELLOW_2:41;
A2: f preserves_inf_of {x,y} by A1;
  reconsider fx = f.x as Element of L;
  thus (sup I1) "/\" (sup I2) = fx "/\" (sup I2) by YELLOW_2:def 3
    .= f.x "/\" f.y by YELLOW_2:def 3
    .= f.(x "/\" y) by A2,YELLOW_3:8
    .= f.(I1 "/\" I2) by YELLOW_4:58
    .= sup (I1 "/\" I2) by YELLOW_2:def 3;
end;
