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 Th25:
  for a being finite Element of PFuncs (V,C) st (for b being
Element of PFuncs (V, C) st b in u holds b tolerates a ) & u "/\" Atom(V, C).a
  [= v holds Atom(V, C).a [= StrongImpl(V, C).(u, v)
proof
  let a be finite Element of PFuncs (V,C);
  assume that
A1: for b be Element of PFuncs (V, C) st b in u holds b tolerates a and
A2: u "/\" Atom(V, C).a [= v;
  reconsider u9 = u, v9 = v, Aa = (Atom(V, C).a) as Element of SubstitutionSet
  (V, C) by SUBSTLAT:def 4;
A3: now
    let c be set;
A4: a in Aa by Th23;
    assume
A5: c in u;
    then
A6: c in u9;
    then reconsider c9 = c as Element of PFuncs (V, C) by SETWISEO:9;
    c9 tolerates a by A1,A5;
    then
    c \/ a in { s1 \/ t1 where s1,t1 is Element of PFuncs (V,C) : s1 in u9
    & t1 in Aa & s1 tolerates t1 } by A5,A4;
    then
A7: c \/ a in u9 ^ Aa by SUBSTLAT:def 3;
    c is finite by A6,Th1;
    then consider b be set such that
A8: b c= c \/ a and
A9: b in mi(u9 ^ Aa) by A7,SUBSTLAT:10;
    b in (the L_meet of SubstLatt (V, C)).(u, Atom(V, C).a) by A9,
SUBSTLAT:def 4;
    then b in u "/\" (Atom (V, C).a) by LATTICES:def 2;
    then consider d be set such that
A10: d in v and
A11: d c= b by A2,Lm9;
    take e = d;
    thus e in v by A10;
    thus e c= c \/ a by A8,A11;
  end;
  now
    let c be set;
    assume
A12: c in Atom(V, C).a;
    then c = a by Th21;
    then consider b be set such that
A13: b in u9 =>> v9 and
A14: b c= c by A3,Th24;
    c in Aa by A12;
    then c is finite by Th1;
    then consider d be set such that
A15: d c= b and
A16: d in mi(u9 =>> v9) by A13,A14,SUBSTLAT:10;
    take e = d;
    thus e in (StrongImpl(V, C).(u, v)) by A16,Def5;
    thus e c= c by A14,A15;
  end;
  hence thesis by Lm8;
end;
