
theorem Th12:
  for V, C being set, a, b being Element of SubstPoset (V, C)
  holds a <= b iff for x being set st x in a ex y being set st y in b & y c= x
proof
  let V, C be set;
  let a, b be Element of SubstPoset (V, C);
  reconsider a9 = a, b9 = b as Element of SubstLatt (V, C);
  reconsider a1 = a, b1 = b as Element of SubstitutionSet (V, C) by
SUBSTLAT:def 4;
A1: a9% = a & b9% = b;
  hereby
    assume a <= b;
    then a9 [= b9 by A1,LATTICE3:7;
    then a9 = a9 "/\" b9 by LATTICES:4
      .= (the L_meet of SubstLatt (V, C)).(a9,b9) by LATTICES:def 2
      .= mi (a1 ^ b1) by SUBSTLAT:def 4;
    hence for x being set st x in a ex y being set st y in b & y c= x by
HEYTING2:4;
  end;
  assume for x being set st x in a ex y being set st y in b & y c= x;
  then mi (a1 ^ b1) = a1 by HEYTING2:5;
  then a9 = (the L_meet of SubstLatt (V, C)).(a9,b9) by SUBSTLAT:def 4
    .= a9 "/\" b9 by LATTICES:def 2;
  then a9 [= b9 by LATTICES:4;
  hence thesis by A1,LATTICE3:7;
end;
