
theorem Th14:
  for L being complete LATTICE for c1,c2 being closure Function of
  L,L holds c1 <= c2 iff Image c2 is SubRelStr of Image c1
proof
  let L be complete LATTICE;
  let c1,c2 be closure Function of L,L;
  hereby
    assume
A1: c1 <= c2;
    the carrier of Image c2 c= the carrier of Image c1
    proof
      let x be object;
      assume x in the carrier of Image c2;
      then consider y being Element of L such that
A2:   c2.y = x by YELLOW_2:10;
A3:   c2.(c2.y) = c2.y by YELLOW_2:18;
A4:   c1.(c2.y) <= c2.(c2.y) by A1,YELLOW_2:9;
      c2.y <= c1.(c2.y) by Th5;
      then c1.(c2.y) = x by A2,A4,A3,ORDERS_2:2;
      then x in rng c1 by FUNCT_2:4;
      hence thesis by YELLOW_0:def 15;
    end;
    hence Image c2 is SubRelStr of Image c1 by Th13;
  end;
  assume that
A5: the carrier of Image c2 c= the carrier of Image c1 and
  the InternalRel of Image c2 c= the InternalRel of Image c1;
  now
    let x be Element of L;
    c2.x in rng c2 by FUNCT_2:4;
    then c2.x in the carrier of Image c2 by YELLOW_0:def 15;
    then ex a being Element of L st c1.a = c2.x by A5,YELLOW_2:10;
    then
A6: c1.(c2.x) = c2.x by YELLOW_2:18;
    x <= c2.x by Th5;
    hence c1.x <= c2.x by A6,WAYBEL_1:def 2;
  end;
  hence thesis by YELLOW_2:9;
end;
