reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;
reserve
  B,A,M for BinOp of D,
  F,G for D* -valued FinSequence,
  f for FinSequence of D,
  d,d1,d2 for Element of D;
reserve
  F,G for non-empty non empty FinSequence of D*,
  f for non empty FinSequence of D;
reserve f,g for FinSequence of D,
        a,b,c for set,
        F,F1,F2 for finite set;

theorem Th88:
  for E be Enumeration of F st X misses union F holds
    ex Ex be Enumeration of UNION(F,{X}) st
      for i st i in dom E holds Ex.i = X\/E.i
proof
  let E be Enumeration of F such that
A1: X misses union F;
  deffunc F(set)=E.$1\/X;
  consider f be Function such that
A2: dom f=dom E & for A be set st A in dom E holds f.A = F(A)
    from FUNCT_1:sch 5;
  dom E=Seg len E by FINSEQ_1:def 3;
  then reconsider f as FinSequence by A2,FINSEQ_1:def 2;
A3: rng f c= UNION(F,{X})
  proof
    let y be object;
    assume y in rng f;
    then consider x such that
A4:   x in dom f & f.x =y by FUNCT_1:def 3;
A5:   E.x in rng E = F by A2,A4,FUNCT_1:def 3,RLAFFIN3:def 1;
A6:   f.x = (E.x) \/X by A4,A2;
    X in {X} by TARSKI:def 1;
    hence thesis by A4,A6,A5,SETFAM_1:def 4;
  end;
  UNION(F,{X}) c= rng f
  proof
    let xy be object;
    assume xy in UNION(F,{X});
    then consider y,x be set such that
A7:   y in F & x in {X} & xy=y\/x by SETFAM_1:def 4;
    y in rng E by A7,RLAFFIN3:def 1;
    then consider z be object such that
A8:   z in dom E & E.z=y by FUNCT_1:def 3;
    x=X by A7,TARSKI:def 1;
    then f.z=xy by A7,A2,A8;
    hence thesis by A2,A8,FUNCT_1:def 3;
  end;
  then
A9: UNION(F,{X}) = rng f by A3;
  f is one-to-one
  proof
    let x,y such that
A10:  x in dom f & y in dom f & f.x=f.y;
    reconsider x,y as set by TARSKI:1;
    E.x in rng E & E.y in rng E by A10,A2,FUNCT_1:def 3;
    then
A11:  E.x misses X & E.y misses X by A1,XBOOLE_1:63,ZFMISC_1:74;
    f.x=F(x) & f.y = F(y) by A10,A2;
    then E.x = E.y by A10,A11,XBOOLE_1:71;
    hence thesis by A2,A10,FUNCT_1:def 4;
  end;
  then reconsider f as Enumeration of UNION(F,{X}) by A9,RLAFFIN3:def 1;
  take f;
  thus thesis by A2;
end;
