reserve V, C, x, a, b for set;
reserve A, B for Element of SubstitutionSet (V, C);
reserve C for finite set;
reserve A, B for Element of SubstitutionSet (V, C);
reserve u, v for Element of SubstLatt (V, C);
reserve s, t, a, b for Element of PFuncs (V,C);
reserve K, L for Element of SubstitutionSet (V, C);

theorem Th27:
  u "/\" StrongImpl(V, C).(u, v) [= v
proof
  now
    reconsider u9 = u, v9 = v as Element of SubstitutionSet (V, C) by
SUBSTLAT:def 4;
    let a be set;
    assume
A1: a in u "/\" StrongImpl(V, C).(u, v);
    u "/\" StrongImpl(V, C).(u, v) = M(V, C).(u, StrongImpl(V, C).(u, v))
    by LATTICES:def 2
      .= M(V, C).(u, mi(u9 =>> v9)) by Def5
      .= mi(u9 ^ mi(u9 =>> v9)) by SUBSTLAT:def 4
      .= mi(u9 ^ (u9 =>> v9)) by SUBSTLAT:20;
    then a in u9 ^ (u9 =>> v9) by A1,SUBSTLAT:6;
    hence ex b be set st b in v & b c= a by Lm3;
  end;
  hence thesis by Lm8;
end;
