
theorem Th39:
  for L being up-complete Semilattice st for I1, I2 being Ideal of
  L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2) holds SupMap L is
  meet-preserving
proof
  let L be up-complete Semilattice such that
A1: for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1
  "/\" I2);
  let x, y be Element of InclPoset(Ids L);
  set f = SupMap L;
  assume ex_inf_of {x,y},InclPoset(Ids L);
  reconsider x1 = x, y1 = y as Ideal of L by YELLOW_2:41;
A2: dom f = the carrier of InclPoset(Ids L) by FUNCT_2:def 1;
  then f.:{x,y} = {f.x,f.y} by FUNCT_1:60;
  hence ex_inf_of f.:{x,y},L by YELLOW_0:21;
  thus inf (f.:{x,y}) = inf {f.x,f.y} by A2,FUNCT_1:60
    .= f.x "/\" f.y by YELLOW_0:40
    .= f.x "/\" (sup y1) by YELLOW_2:def 3
    .= (sup x1) "/\" (sup y1) by YELLOW_2:def 3
    .= sup (x1 "/\" y1) by A1
    .= f.(x1 "/\" y1) by YELLOW_2:def 3
    .= f.(x "/\" y) by YELLOW_4:58
    .= f.inf {x,y} by YELLOW_0:40;
end;
