reserve A for non empty set,
  a for Element of A;
reserve A for set;
reserve B,C for Element of Fin DISJOINT_PAIRS A,
  x for Element of [:Fin A, Fin A:],
  a,b,c,d,s,t,s9,t9,t1,t2,s1,s2 for Element of DISJOINT_PAIRS A,
  u,v,w for Element of NormForm A;
reserve K,L for Element of Normal_forms_on A;
reserve f,f9 for (Element of Funcs(DISJOINT_PAIRS A, [:Fin A,Fin A:])),
  g,h for Element of Funcs(DISJOINT_PAIRS A, [A]);

theorem Th26:
  (for b st b in u holds b \/ a in DISJOINT_PAIRS A ) & u "/\"
  Atom(A).a [= w implies Atom(A).a [= StrongImpl(A).(u, w)
proof
  assume that
A1: for b st b in u holds b \/ a in DISJOINT_PAIRS A and
A2: u "/\" Atom(A).a [= w;
A3: now
    let c;
    assume
A4: c in u;
    then
A5: c \/ a is Element of DISJOINT_PAIRS A by A1;
    a in @(Atom(A).a) by Th7;
    then c \/ a in @u ^ @(Atom(A).a) by A1,A4,NORMFORM:35;
    then consider b such that
A6: b c= c \/ a and
A7: b in mi(@u ^ @(Atom(A).a)) by A5,NORMFORM:41;
    b in M(A).(u, Atom(A).a) by A7,NORMFORM:def 12;
    then b in u "/\" Atom(A).a by LATTICES:def 2;
    then consider d such that
A8: d in w and
A9: d c= b by A2,Lm2;
    take e = d;
    thus e in w by A8;
    thus e c= c \/ a by A6,A9,NORMFORM:2;
  end;
  now
    let c;
    assume c in Atom(A).a;
    then c = a by Th6;
    then consider b such that
A10: b in @u =>> @w and
A11: b c= c by A3,Th20;
    consider d such that
A12: d c= b and
A13: d in mi(@u =>> @w) by A10,NORMFORM:41;
    take e = d;
    thus e in (StrongImpl(A).(u, w)) by A13,Def9;
    thus e c= c by A11,A12,NORMFORM:2;
  end;
  hence thesis by Lm3;
end;
