A7: dom F₃() = omega by FUNCT_2:def 1;
A8: F₂() in rng F₃() by A5, A7, FUNCT_1:def 3;
hence F₂() c= Union F₃() by ZFMISC_1:74; :: thesis: ( F₄((Union F₃())) = Union F₃() & ( for b being ordinal number st F₂() c= b & b in F₁() & F₄(b) = b holds
Union F₃() c= b ) )
set phi = F₃();

A9: now :: thesis: for a being ordinal number holds not S₁[a]

defpred S₁[ Ordinal] means ( $1 in F₁() & $1 c/= F₄($1) );
assume A10: ex a being ordinal number st S₁[a] ; :: thesis: contradiction
consider a being ordinal number such that
A11: S₁[a] and
A12: for b being ordinal number st S₁[b] holds
a c= b from ORDINAL1:sch 1(A10);
F₄(F₄(a)) in F₄(a) by A3, A11, ORDINAL1:16;
then F₄(a) c/= F₄(F₄(a)) by ORDINAL1:5;
hence contradiction by A2, A11, A12; :: thesis: verum

end;

then F₂() c= F₄(F₂()) ;
then A13: ( F₂() c< F₄(F₂()) or F₂() = F₄(F₂()) ) by XBOOLE_0:def 8;

per cases ( F₄(F₂()) = F₂() or F₂() in F₄(F₂()) ) by A13, ORDINAL1:11;

suppose A14: F₄(F₂()) = F₂() ; :: thesis: ( F₄((Union F₃())) = Union F₃() & ( for b being ordinal number st F₂() c= b & b in F₁() & F₄(b) = b holds
Union F₃() c= b ) )

A15: for a being set st a in omega holds
F₃() . a = F₂()

proof

let a be set ; :: thesis: ( a in omega implies F₃() . a = F₂() )
assume a in omega ; :: thesis: F₃() . a = F₂()
then reconsider a = a as Element of omega ;
defpred S₁[ Nat] means F₃() . $1 = F₂();
A16: S₁[ 0 ] by A5;

A17: now :: thesis: for n being Nat st S₁[n] holds
S₁[n + 1]

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A18: S₁[n] ; :: thesis: S₁[n + 1]
n in omega by ORDINAL1:def 12;
then F₃() . (succ n) = F₄(F₂()) by A6, A18;
hence S₁[n + 1] by A14, NAT_1:38; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A16, A17);
then S₁[a] ;
hence F₃() . a = F₂() ; :: thesis: verum

end;

rng F₃() = {F₂()}

proof

thus rng F₃() c= {F₂()} :: according to XBOOLE_0:def 10 :: thesis: {F₂()} c= rng F₃()

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng F₃() or x in {F₂()} )
assume x in rng F₃() ; :: thesis: x in {F₂()}
then consider a being set such that
A19: ( a in dom F₃() & x = F₃() . a ) by FUNCT_1:def 3;
x = F₂() by A15, A19, A7;
hence x in {F₂()} by TARSKI:def 1; :: thesis: verum

end;

thus {F₂()} c= rng F₃() by A8, ZFMISC_1:31; :: thesis: verum

end;

then Union F₃() = F₂() by ZFMISC_1:25;
hence ( F₄((Union F₃())) = Union F₃() & ( for b being ordinal number st F₂() c= b & b in F₁() & F₄(b) = b holds
Union F₃() c= b ) ) by A14; :: thesis: verum

end;

suppose A20: F₂() in F₄(F₂()) ; :: thesis: ( F₄((Union F₃())) = Union F₃() & ( for b being ordinal number st F₂() c= b & b in F₁() & F₄(b) = b holds
Union F₃() c= b ) )

deffunc H₁( Ordinal, Ordinal) -> Ordinal = F₄($2);
deffunc H₂( Ordinal, T-Sequence) -> set = {} ;

A21: now :: thesis: for a being ordinal number st succ a in omega holds
F₃() . (succ a) = H₁(a,F₃() . a)

let a be ordinal number ; :: thesis: ( succ a in omega implies F₃() . (succ a) = H₁(a,F₃() . a) )
assume A22: succ a in omega ; :: thesis: F₃() . (succ a) = H₁(a,F₃() . a)
a in succ a by ORDINAL1:6;
hence F₃() . (succ a) = H₁(a,F₃() . a) by A6, A22, ORDINAL1:10; :: thesis: verum

end;

A23: for a being ordinal number st a in omega holds
( F₂() c= F₃() . a & F₃() . a in F₃() . (succ a) )

proof

let a be ordinal number ; :: thesis: ( a in omega implies ( F₂() c= F₃() . a & F₃() . a in F₃() . (succ a) ) )
assume a in omega ; :: thesis: ( F₂() c= F₃() . a & F₃() . a in F₃() . (succ a) )
then reconsider a = a as Element of omega ;
defpred S₁[ Nat] means ( F₂() c= F₃() . $1 & F₃() . $1 in F₃() . (succ $1) );
A24: S₁[ 0 ] by A20, A5, A6;

A25: now :: thesis: for n being Nat st S₁[n] holds
S₁[n + 1]

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A26: S₁[n] ; :: thesis: S₁[n + 1]
A27: ( n + 1 = succ n & (n + 1) + 1 = succ (n + 1) & n + 1 in omega & (n + 1) + 1 in omega ) by NAT_1:38;
then ( F₃() . (n + 1) = F₄((F₃() . n)) & F₃() . ((n + 1) + 1) = F₄((F₃() . (n + 1))) ) by A21;
then ( F₃() . n c= F₃() . (n + 1) & F₃() . (n + 1) in F₃() . ((n + 1) + 1) ) by A3, A9, A26, A27;
hence S₁[n + 1] by A26, NAT_1:38, XBOOLE_1:1; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A24, A25);
then S₁[a] ;
hence ( F₂() c= F₃() . a & F₃() . a in F₃() . (succ a) ) ; :: thesis: verum

end;

deffunc H₃( Ordinal) -> Ordinal = F₄($1);
consider fi being Ordinal-Sequence such that
A28: ( dom fi = Union F₃() & ( for a being ordinal number st a in Union F₃() holds
fi . a = H₃(a) ) ) from ORDINAL2:sch 3();
1 = succ 0 ;
then F₃() . 1 = H₃(F₂()) by A5, A6;
then H₃(F₂()) in rng F₃() by A7, FUNCT_1:def 3;
then A29: H₃(F₂()) c= Union F₃() by ZFMISC_1:74;
A30: Union F₃() in F₁() by A1, Th41;

now :: thesis: for c being ordinal number st c in Union F₃() holds
succ c in Union F₃()

let c be ordinal number ; :: thesis: ( c in Union F₃() implies succ c in Union F₃() )
assume c in Union F₃() ; :: thesis: succ c in Union F₃()
then consider x being set such that
A31: ( x in dom F₃() & c in F₃() . x ) by CARD_5:2;
reconsider x = x as Element of omega by A31, FUNCT_2:def 1;
( succ c c= F₃() . x & F₃() . x in F₃() . (succ x) ) by A23, A31, ORDINAL1:21;
then ( succ c in F₃() . (succ x) & succ x in omega ) by ORDINAL1:12;
hence succ c in Union F₃() by A7, CARD_5:2; :: thesis: verum

end;

then A32: Union F₃() is limit_ordinal by ORDINAL1:28;
then A33: H₃( Union F₃()) is_limes_of fi by A4, A20, A28, A29, A30;
fi is increasing

proof

let a be ordinal number ; :: according to ORDINAL2:def 12 :: thesis: for b₁ being set holds
( not a in b₁ or not b₁ in dom fi or fi . a in fi . b₁ )
let b be ordinal number ; :: thesis: ( not a in b or not b in dom fi or fi . a in fi . b )
assume A34: ( a in b & b in dom fi ) ; :: thesis: fi . a in fi . b
then ( fi . a = H₃(a) & fi . b = H₃(b) & b in F₁() ) by A30, A28, ORDINAL1:10;
hence fi . a in fi . b by A3, A34; :: thesis: verum

end;

then A35: sup fi = lim fi by A20, A28, A29, A32, ORDINAL4:8
.= H₃( Union F₃()) by A33, ORDINAL2:def 10 ;
thus H₃( Union F₃()) c= Union F₃() :: according to XBOOLE_0:def 10 :: thesis: ( Union F₃() c= F₄((Union F₃())) & ( for b being ordinal number st F₂() c= b & b in F₁() & F₄(b) = b holds
Union F₃() c= b ) )

proof

let x be Ordinal; :: according to ORDINAL1:def 5 :: thesis: ( not x in H₃( Union F₃()) or x in Union F₃() )
assume A36: x in H₃( Union F₃()) ; :: thesis: x in Union F₃()
reconsider A = x as Ordinal ;
consider b being ordinal number such that
A37: ( b in rng fi & A c= b ) by A35, A36, ORDINAL2:21;
consider y being set such that
A38: ( y in dom fi & b = fi . y ) by A37, FUNCT_1:def 3;
reconsider y = y as Ordinal by A38;
consider z being set such that
A39: ( z in dom F₃() & y in F₃() . z ) by A28, A38, CARD_5:2;
reconsider z = z as Ordinal by A39;
set c = F₃() . z;
succ z in omega by A7, A39, ORDINAL1:28;
then A40: ( F₃() . (succ z) = H₃(F₃() . z) & F₃() . (succ z) in rng F₃() & b = H₃(y) ) by A7, A21, A28, A38, FUNCT_1:def 3;
( b in H₃(F₃() . z) & H₃(F₃() . z) c= Union F₃() ) by A40, A3, A39, ZFMISC_1:74;
hence x in Union F₃() by A37, ORDINAL1:12; :: thesis: verum

end;

thus Union F₃() c= H₃( Union F₃()) by A9, A1, Th41; :: thesis: for b being ordinal number st F₂() c= b & b in F₁() & F₄(b) = b holds
Union F₃() c= b
let b be ordinal number ; :: thesis: ( F₂() c= b & b in F₁() & F₄(b) = b implies Union F₃() c= b )
assume A41: ( F₂() c= b & b in F₁() & H₃(b) = b ) ; :: thesis: Union F₃() c= b
rng F₃() c= b

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng F₃() or x in b )
assume x in rng F₃() ; :: thesis: x in b
then consider a being set such that
A42: ( a in dom F₃() & x = F₃() . a ) by FUNCT_1:def 3;
reconsider a = a as Element of omega by A42, FUNCT_2:def 1;
defpred S₁[ Nat] means F₃() . $1 in b;
F₂() <> b by A20, A41;
then F₂() c< b by A41, XBOOLE_0:def 8;
then A43: S₁[ 0 ] by A5, ORDINAL1:11;

A44: now :: thesis: for n being Nat st S₁[n] holds
S₁[n + 1]

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume S₁[n] ; :: thesis: S₁[n + 1]
then ( H₃(F₃() . n) in b & n in omega ) by A3, A41, ORDINAL1:def 12;
then F₃() . (succ n) in b by A6;
hence S₁[n + 1] by NAT_1:38; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A43, A44);
then S₁[a] ;
hence x in b by A42; :: thesis: verum

end;

then ( Union F₃() c= union b & union b c= b ) by ORDINAL2:5, ZFMISC_1:77;
hence Union F₃() c= b by XBOOLE_1:1; :: thesis: verum

end;

end;