
theorem Th12:
  for L being complete LATTICE for c1,c2 being Function of L,L for
  x,y being Element of ClOpers L st x = c1 & y = c2 holds x <= y iff c1 <= c2
proof
  let L be complete LATTICE;
  let c1,c2 be Function of L,L, x,y be Element of ClOpers L such that
A1: x = c1 and
A2: y = c2;
  reconsider x9 = x, y9 = y as Element of MonMaps(L,L) by YELLOW_0:58;
  reconsider x99 = x9, y99 = y9 as Element of L|^the carrier of L by
YELLOW_0:58;
  x99 <= y99 iff x9 <= y9 by YELLOW_0:59,60;
  hence thesis by A1,A2,Th11,YELLOW_0:59,60;
end;
