F0: F₁() in rng F₂() by F1, FUNCT_1:def 5;
hence F₁() c= Union F₂() by ZFMISC_1:92; :: thesis: ( F₃((Union F₂())) = Union F₂() & ( for b being ordinal number st F₁() c= b & F₃(b) = b holds
Union F₂() c= b ) )
set phi = F₂();

A7: now

defpred S₁[ Ordinal] means not $1 c= F₃($1);
assume A8: ex a being ordinal number st S₁[a] ; :: thesis: contradiction
consider a being ordinal number such that
A9: S₁[a] and
A10: for b being ordinal number st S₁[b] holds
a c= b from ORDINAL1:sch 1(A8);
F₃(F₃(a)) in F₃(a) by A1, A9, ORDINAL1:26;
then not F₃(a) c= F₃(F₃(a)) by ORDINAL1:7;
hence contradiction by A9, A10; :: thesis: verum

end;

then F₁() c= F₃(F₁()) ;
then F3: ( F₁() c< F₃(F₁()) or F₁() = F₃(F₁()) ) by XBOOLE_0:def 8;

per cases ( F₃(F₁()) = F₁() or F₁() in F₃(F₁()) ) by F3, ORDINAL1:21;

suppose A3: F₃(F₁()) = F₁() ; :: thesis: ( F₃((Union F₂())) = Union F₂() & ( for b being ordinal number st F₁() c= b & F₃(b) = b holds
Union F₂() c= b ) )

A4: for a being set st a in omega holds
F₂() . a = F₁()

proof

let a be set ; :: thesis: ( a in omega implies F₂() . a = F₁() )
assume a in omega ; :: thesis: F₂() . a = F₁()
then reconsider a = a as Element of omega ;
defpred S₁[ Nat] means F₂() . $1 = F₁();
C0: S₁[ 0 ] by F1;

C1: now

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume C2: S₁[n] ; :: thesis: S₁[n + 1]
n in omega by ORDINAL1:def 13;
then F₂() . (succ n) = F₃(F₁()) by F2, C2;
hence S₁[n + 1] by A3, NAT_1:39; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(C0, C1);
then S₁[a] ;
hence F₂() . a = F₁() ; :: thesis: verum

end;

rng F₂() = {F₁()}

proof

thus rng F₂() c= {F₁()} :: according to XBOOLE_0:def 10 :: thesis: {F₁()} c= rng F₂()

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng F₂() or x in {F₁()} )
assume x in rng F₂() ; :: thesis: x in {F₁()}
then consider a being set such that
G1: ( a in dom F₂() & x = F₂() . a ) by FUNCT_1:def 5;
x = F₁() by A4, G1, F1;
hence x in {F₁()} by TARSKI:def 1; :: thesis: verum

end;

thus {F₁()} c= rng F₂() by F0, ZFMISC_1:37; :: thesis: verum

end;

then Union F₂() = F₁() by ZFMISC_1:31;
hence ( F₃((Union F₂())) = Union F₂() & ( for b being ordinal number st F₁() c= b & F₃(b) = b holds
Union F₂() c= b ) ) by A3; :: thesis: verum

end;

suppose A3: F₁() in F₃(F₁()) ; :: thesis: ( F₃((Union F₂())) = Union F₂() & ( for b being ordinal number st F₁() c= b & F₃(b) = b holds
Union F₂() c= b ) )

deffunc H₁( Ordinal, Ordinal) -> Ordinal = F₃($2);
deffunc H₂( Ordinal, T-Sequence) -> set = {} ;

A6: now

let a be ordinal number ; :: thesis: ( succ a in omega implies F₂() . (succ a) = H₁(a,F₂() . a) )
assume G1: succ a in omega ; :: thesis: F₂() . (succ a) = H₁(a,F₂() . a)
a in succ a by ORDINAL1:10;
hence F₂() . (succ a) = H₁(a,F₂() . a) by F2, G1, ORDINAL1:19; :: thesis: verum

end;

B1: for a being ordinal number st a in omega holds
( F₁() c= F₂() . a & F₂() . a in F₂() . (succ a) )

proof

let a be ordinal number ; :: thesis: ( a in omega implies ( F₁() c= F₂() . a & F₂() . a in F₂() . (succ a) ) )
assume a in omega ; :: thesis: ( F₁() c= F₂() . a & F₂() . a in F₂() . (succ a) )
then reconsider a = a as Element of omega ;
defpred S₁[ Nat] means ( F₁() c= F₂() . $1 & F₂() . $1 in F₂() . (succ $1) );
B01: S₁[ 0 ] by A3, F1, F2;

B02: now

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume B03: S₁[n] ; :: thesis: S₁[n + 1]
B04: ( n + 1 = succ n & (n + 1) + 1 = succ (n + 1) & n + 1 in omega & (n + 1) + 1 in omega ) by NAT_1:39;
then ( F₂() . (n + 1) = F₃((F₂() . n)) & F₂() . ((n + 1) + 1) = F₃((F₂() . (n + 1))) ) by A6;
then ( F₂() . n c= F₂() . (n + 1) & F₂() . (n + 1) in F₂() . ((n + 1) + 1) ) by A1, A7, B03, B04;
hence S₁[n + 1] by B03, B04, XBOOLE_1:1; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(B01, B02);
then S₁[a] ;
hence ( F₁() c= F₂() . a & F₂() . a in F₂() . (succ a) ) ; :: thesis: verum

end;

deffunc H₃( Ordinal) -> Ordinal = F₃($1);
consider fi being Ordinal-Sequence such that
A23: ( dom fi = Union F₂() & ( for a being ordinal number st a in Union F₂() holds
fi . a = H₃(a) ) ) from ORDINAL2:sch 3();
1 = succ 0 ;
then F₂() . 1 = H₃(F₁()) by F1, F2;
then H₃(F₁()) in rng F₂() by F1, FUNCT_1:def 5;
then H₃(F₁()) c= Union F₂() by ZFMISC_1:92;
then A24: Union F₂() <> {} by A3;

now

let c be ordinal number ; :: thesis: ( c in Union F₂() implies succ c in Union F₂() )
assume c in Union F₂() ; :: thesis: succ c in Union F₂()
then consider x being set such that
V1: ( x in dom F₂() & c in F₂() . x ) by CARD_5:10;
reconsider x = x as Element of omega by F1, V1;
( succ c c= F₂() . x & F₂() . x in F₂() . (succ x) ) by B1, V1, ORDINAL1:33;
then ( succ c in F₂() . (succ x) & succ x in omega ) by ORDINAL1:22, ORDINAL1:def 13;
hence succ c in Union F₂() by F1, CARD_5:10; :: thesis: verum

end;

then A249: Union F₂() is limit_ordinal by ORDINAL1:41;
then A25: H₃( Union F₂()) is_limes_of fi by A2, A23, A24;
fi is increasing

proof

let a be ordinal number ; :: according to ORDINAL2:def 16 :: thesis: for b₁ being set holds
( not a in b₁ or not b₁ in proj1 fi or fi . a in fi . b₁ )
let b be ordinal number ; :: thesis: ( not a in b or not b in proj1 fi or fi . a in fi . b )
assume A26: ( a in b & b in dom fi ) ; :: thesis: fi . a in fi . b
then ( fi . a = H₃(a) & fi . b = H₃(b) ) by A23, ORDINAL1:19;
hence fi . a in fi . b by A1, A26; :: thesis: verum

end;

then A27: sup fi = lim fi by A23, A24, A249, ORDINAL4:8
.= H₃( Union F₂()) by A25, ORDINAL2:def 14 ;
thus H₃( Union F₂()) c= Union F₂() :: according to XBOOLE_0:def 10 :: thesis: ( Union F₂() c= F₃((Union F₂())) & ( for b being ordinal number st F₁() c= b & F₃(b) = b holds
Union F₂() c= b ) )

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in H₃( Union F₂()) or x in Union F₂() )
assume A28: x in H₃( Union F₂()) ; :: thesis: x in Union F₂()
then reconsider A = x as Ordinal ;
consider b being ordinal number such that
A29: ( b in rng fi & A c= b ) by A27, A28, ORDINAL2:29;
consider y being set such that
A30: ( y in dom fi & b = fi . y ) by A29, FUNCT_1:def 5;
reconsider y = y as Ordinal by A30;
consider z being set such that
A32: ( z in dom F₂() & y in F₂() . z ) by A23, A30, CARD_5:10;
reconsider z = z as Ordinal by A32;
set c = F₂() . z;
succ z in omega by F1, A32, ORDINAL1:41;
then ( F₂() . (succ z) = H₃(F₂() . z) & F₂() . (succ z) in rng F₂() & b = H₃(y) ) by F1, A6, A23, A30, FUNCT_1:def 5;
then ( b in H₃(F₂() . z) & H₃(F₂() . z) c= Union F₂() ) by ZFMISC_1:92, A1, A32;
hence x in Union F₂() by A29, ORDINAL1:22; :: thesis: verum

end;

thus Union F₂() c= H₃( Union F₂()) by A7; :: thesis: for b being ordinal number st F₁() c= b & F₃(b) = b holds
Union F₂() c= b
let b be ordinal number ; :: thesis: ( F₁() c= b & F₃(b) = b implies Union F₂() c= b )
assume Z1: ( F₁() c= b & H₃(b) = b ) ; :: thesis: Union F₂() c= b
rng F₂() c= b

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng F₂() or x in b )
assume x in rng F₂() ; :: thesis: x in b
then consider a being set such that
Z2: ( a in dom F₂() & x = F₂() . a ) by FUNCT_1:def 5;
reconsider a = a as Element of omega by F1, Z2;
defpred S₁[ Nat] means F₂() . $1 in b;
F₁() <> b by A3, Z1;
then F₁() c< b by Z1, XBOOLE_0:def 8;
then Z3: S₁[ 0 ] by F1, ORDINAL1:21;

Z4: now

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume S₁[n] ; :: thesis: S₁[n + 1]
then ( H₃(F₂() . n) in b & n in omega ) by A1, Z1, ORDINAL1:def 13;
then F₂() . (succ n) in b by F2;
hence S₁[n + 1] by NAT_1:39; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(Z3, Z4);
then S₁[a] ;
hence x in b by Z2; :: thesis: verum

end;

then ( Union F₂() c= union b & union b c= b ) by ZFMISC_1:95, ORDINAL2:5;
hence Union F₂() c= b by XBOOLE_1:1; :: thesis: verum

end;

end;