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;

theorem Th9:
  FinJoin(K, Atom(A)) = FinUnion(K, singleton DISJOINT_PAIRS A)
proof
  deffunc F(Element of Fin DISJOINT_PAIRS A) = mi $1;
A1: FinUnion(K,singleton DISJOINT_PAIRS A) c= mi (FinUnion(K,singleton
  DISJOINT_PAIRS A))
  proof
    let a;
    assume
A2: a in FinUnion(K,singleton DISJOINT_PAIRS A);
    then consider b such that
A3: b in K and
A4: a in (singleton DISJOINT_PAIRS A).b by SETWISEO:57;
A5: a = b by A4,SETWISEO:55;
    now
      let s;
      assume that
A6:   s in FinUnion(K,singleton DISJOINT_PAIRS A) and
A7:   s c= a;
      consider t such that
A8:  t in K and
A9:  s in (singleton DISJOINT_PAIRS A).t by A6,SETWISEO:57;
      s = t by A9,SETWISEO:55;
      hence s = a by A3,A5,A7,A8,NORMFORM:32;
    end;
    hence thesis by A2,NORMFORM:39;
  end;
A10: mi (FinUnion(K,singleton DISJOINT_PAIRS A)) c= FinUnion(K,singleton
  DISJOINT_PAIRS A) by NORMFORM:40;
  consider g being Function of Fin DISJOINT_PAIRS A, Normal_forms_on A such
  that
A11: g.B = F(B) from FUNCT_2:sch 4;
  reconsider g as Function of Fin DISJOINT_PAIRS A,the carrier of NormForm A
  by NORMFORM:def 12;
A12: g.{}.DISJOINT_PAIRS A = mi {}.DISJOINT_PAIRS A by A11
    .= {} by NORMFORM:40,XBOOLE_1:3
    .= Bottom NormForm A by NORMFORM:57
    .= the_unity_wrt the L_join of NormForm A by LATTICE2:18;
A13: now
    let x,y be Element of Fin DISJOINT_PAIRS A;
A14: @(g.x) = mi x & @(g.y) = mi y by A11;
    thus g.(x \/ y) = mi (x \/ y) by A11
      .= mi (mi x \/ y) by NORMFORM:44
      .= mi (mi x \/ mi y) by NORMFORM:45
      .= (the L_join of NormForm A).(g.x,g.y) by A14,NORMFORM:def 12;
  end;
A15: now
    let a;
    thus (g*singleton DISJOINT_PAIRS A).a = g.(singleton DISJOINT_PAIRS A .a)
    by FUNCT_2:15
      .= g.{a} by SETWISEO:54
      .= mi { a } by A11
      .= { a } by NORMFORM:42
      .= Atom A.a by Def4;
  end;
  thus FinJoin(K, Atom A) = (the L_join of NormForm A) $$(K,Atom A) by
LATTICE2:def 3
    .= (the L_join of NormForm A) $$(K,g*singleton DISJOINT_PAIRS A) by A15,
FUNCT_2:63
    .= g.(FinUnion(K,singleton DISJOINT_PAIRS A)) by A12,A13,SETWISEO:53
    .= mi (FinUnion(K,singleton DISJOINT_PAIRS A)) by A11
    .= FinUnion(K,singleton DISJOINT_PAIRS A) by A10,A1;
end;
