
theorem Th60:
  for S being Semilattice, x be Element of S holds x "/\" is monotone
proof
  let S be Semilattice, x be Element of S;
  let s1,s2 be Element of S;
  assume
A1: s1 <= s2;
A2: ex_inf_of {x,s1},S by YELLOW_0:21;
  then
A3: x"/\"s1 <= x by YELLOW_0:19;
  x"/\"s1 <= s1 by A2,YELLOW_0:19;
  then ex_inf_of {x,s2},S & x"/\"s1 <= s2 by A1,ORDERS_2:3,YELLOW_0:21;
  then x"/\"s1 <= x"/\"s2 by A3,YELLOW_0:19;
  then (x "/\").s1 <= x"/\"s2 by Def18;
  hence (x "/\").s1 <= (x "/\").s2 by Def18;
end;
