F1a: dom F₃() = omega by FUNCT_2:def 1;
F0: F₂() in rng F₃() by F1, F1a, 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();

A7: now

defpred S₁[ Ordinal] means ( $1 in F₁() & $1 c/= F₄($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);
F₄(F₄(a)) in F₄(a) by A1, A9, ORDINAL1:16;
then F₄(a) c/= F₄(F₄(a)) by ORDINAL1:5;
hence contradiction by P0, A9, A10; :: thesis: verum

end;

then F₂() c= F₄(F₂()) ;
then F3: ( F₂() c< F₄(F₂()) or F₂() = F₄(F₂()) ) by XBOOLE_0:def 8;

per cases ( F₄(F₂()) = F₂() or F₂() in F₄(F₂()) ) by F3, ORDINAL1:11;

suppose A3: 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 ) )

A4: 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();
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 12;
then F₃() . (succ n) = F₄(F₂()) by F2, C2;
hence S₁[n + 1] by A3, NAT_1:38; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(C0, C1);
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
G1: ( a in dom F₃() & x = F₃() . a ) by FUNCT_1:def 3;
x = F₂() by A4, G1, F1a;
hence x in {F₂()} by TARSKI:def 1; :: thesis: verum

end;

thus {F₂()} c= rng F₃() by F0, 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 A3; :: thesis: verum

end;

suppose A3: 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 = {} ;

A6: now

let a be ordinal number ; :: thesis: ( succ a in omega implies F₃() . (succ a) = H₁(a,F₃() . a) )
assume G1: 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 F2, G1, ORDINAL1:10; :: thesis: verum

end;

B1: 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) );
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_1:38;
then ( F₃() . (n + 1) = F₄((F₃() . n)) & F₃() . ((n + 1) + 1) = F₄((F₃() . (n + 1))) ) by A6;
then ( F₃() . n c= F₃() . (n + 1) & F₃() . (n + 1) in F₃() . ((n + 1) + 1) ) by A1, A7, B03, B04;
hence S₁[n + 1] by B03, NAT_1:38, XBOOLE_1:1; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(B01, B02);
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
A23: ( 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 F1, F2;
then H₃(F₂()) in rng F₃() by F1a, FUNCT_1:def 3;
then A24: H₃(F₂()) c= Union F₃() by ZFMISC_1:74;
J1: Union F₃() in F₁() by A0, ThC1;

now

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
V1: ( x in dom F₃() & c in F₃() . x ) by CARD_5:2;
reconsider x = x as Element of omega by V1, FUNCT_2:def 1;
( succ c c= F₃() . x & F₃() . x in F₃() . (succ x) ) by B1, V1, ORDINAL1:21;
then ( succ c in F₃() . (succ x) & succ x in omega ) by ORDINAL1:12;
hence succ c in Union F₃() by F1a, CARD_5:2; :: thesis: verum

end;

then A249: Union F₃() is limit_ordinal by ORDINAL1:28;
then A25: H₃( Union F₃()) is_limes_of fi by A2, A3, A23, A24, J1;
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 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) & b in F₁() ) by J1, A23, ORDINAL1:10;
hence fi . a in fi . b by A1, A26; :: thesis: verum

end;

then A27: sup fi = lim fi by A3, A23, A24, A249, ORDINAL4:8
.= H₃( Union F₃()) by A25, 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 A28: x in H₃( Union F₃()) ; :: thesis: x in Union F₃()
reconsider A = x as Ordinal ;
consider b being ordinal number such that
A29: ( b in rng fi & A c= b ) by A27, A28, ORDINAL2:21;
consider y being set such that
A30: ( y in dom fi & b = fi . y ) by A29, FUNCT_1:def 3;
reconsider y = y as Ordinal by A30;
consider z being set such that
A32: ( z in dom F₃() & y in F₃() . z ) by A23, A30, CARD_5:2;
reconsider z = z as Ordinal by A32;
set c = F₃() . z;
succ z in omega by F1a, A32, ORDINAL1:28;
then A33: ( F₃() . (succ z) = H₃(F₃() . z) & F₃() . (succ z) in rng F₃() & b = H₃(y) ) by F1a, A6, A23, A30, FUNCT_1:def 3;
( b in H₃(F₃() . z) & H₃(F₃() . z) c= Union F₃() ) by A33, A1, A32, ZFMISC_1:74;
hence x in Union F₃() by A29, ORDINAL1:12; :: thesis: verum

end;

thus Union F₃() c= H₃( Union F₃()) by A7, A0, ThC1; :: 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 Z1: ( 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
Z2: ( a in dom F₃() & x = F₃() . a ) by FUNCT_1:def 3;
reconsider a = a as Element of omega by Z2, FUNCT_2:def 1;
defpred S₁[ Nat] means F₃() . $1 in b;
F₂() <> b by A3, Z1;
then F₂() c< b by Z1, XBOOLE_0:def 8;
then Z3: S₁[ 0 ] by F1, ORDINAL1:11;

Z4: now

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 A1, Z1, ORDINAL1:def 12;
then F₃() . (succ n) in b by F2;
hence S₁[n + 1] by NAT_1:38; :: 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 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;