:: STIRL2_1 semantic presentation

begin


theorem Th1: :: STIRL2_1:1
for N being non empty Subset of NAT holds min N = min* N
proof
let N be non empty Subset of NAT; ::_thesis: min N = min* N
A1: for n being ext-real number st n in N holds
min* N <= n by NAT_1:def_1;
 min* N in N by NAT_1:def_1;
hence  min N = min* N by A1, XXREAL_2:def_7; ::_thesis: verum
end;

theorem Th2: :: STIRL2_1:2
for K, N being non empty Subset of NAT holds min ((min K),(min N)) = min (K \/ N)
proof
let K, N be non empty Subset of NAT; ::_thesis: min ((min K),(min N)) = min (K \/ N)
set m = min ((min N),(min K));
A1: for k being ext-real number st k in N \/ K holds
min ((min N),(min K)) <= k
proof
let k be ext-real number ; ::_thesis: ( k in N \/ K implies min ((min N),(min K)) <= k )
assume  k in N \/ K ; ::_thesis: min ((min N),(min K)) <= k
then  ( k in N or k in K ) by XBOOLE_0:def_3;
then A2: ( min N <= k or min K <= k ) by XXREAL_2:def_7;
A3: min ((min N),(min K)) <= min K by XXREAL_0:17;
 min ((min N),(min K)) <= min N by XXREAL_0:17;
hence  min ((min N),(min K)) <= k by A2, A3, XXREAL_0:2; ::_thesis: verum
end;
 ( min ((min N),(min K)) = min N or min ((min N),(min K)) = min K ) by XXREAL_0:15;
then  ( min ((min N),(min K)) in N or min ((min N),(min K)) in K ) by XXREAL_2:def_7;
then  min ((min N),(min K)) in N \/ K by XBOOLE_0:def_3;
hence  min ((min K),(min N)) = min (K \/ N) by A1, XXREAL_2:def_7; ::_thesis: verum
end;

theorem :: STIRL2_1:3
for Ke, Ne being Subset of NAT holds min ((min* Ke),(min* Ne)) <= min* (Ke \/ Ne)
proof
let Ke, Ne be Subset of NAT; ::_thesis: min ((min* Ke),(min* Ne)) <= min* (Ke \/ Ne)
now__::_thesis:_min_((min*_Ke),(min*_Ne))_<=_min*_(Ke_\/_Ne)
percases ( Ke is empty or Ne is empty or ( not Ke is empty & not Ne is empty ) ) ;
suppose ( Ke is empty or Ne is empty ) ; ::_thesis: min ((min* Ke),(min* Ne)) <= min* (Ke \/ Ne)
then  ( ( min* Ke = 0 & min* Ne >= 0 ) or ( min* Ne = 0 & min* Ke >= 0 ) ) by NAT_1:def_1;
hence  min ((min* Ke),(min* Ne)) <= min* (Ke \/ Ne) by XXREAL_0:def_9; ::_thesis: verum
end;
suppose ( not Ke is empty & not Ne is empty ) ; ::_thesis: min ((min* Ke),(min* Ne)) <= min* (Ke \/ Ne)
then reconsider K = Ke, N = Ne as non empty Subset of NAT ;
A1: min N = min* Ne by Th1;
A2: min (K \/ N) = min* (Ke \/ Ne) by Th1;
 min K = min* Ke by Th1;
hence  min ((min* Ke),(min* Ne)) <= min* (Ke \/ Ne) by A1, A2, Th2; ::_thesis: verum
end;
end;
end;
hence  min ((min* Ke),(min* Ne)) <= min* (Ke \/ Ne) ; ::_thesis: verum
end;

theorem :: STIRL2_1:4
for Ne, Ke being Subset of NAT st not min* Ne in Ne /\ Ke holds
min* Ne = min* (Ne \ Ke)
proof
let Ne, Ke be Subset of NAT; ::_thesis: ( not min* Ne in Ne /\ Ke implies min* Ne = min* (Ne \ Ke) )
assume A1: not min* Ne in Ne /\ Ke ; ::_thesis: min* Ne = min* (Ne \ Ke)
now__::_thesis:_min*_Ne_=_min*_(Ne_\_Ke)
percases ( Ne is empty or not Ne is empty ) ;
suppose Ne is empty ; ::_thesis: min* Ne = min* (Ne \ Ke)
hence  min* Ne = min* (Ne \ Ke) ; ::_thesis: verum
end;
suppose not Ne is empty ; ::_thesis: min* Ne = min* (Ne \ Ke)
then A2: min* Ne in Ne by NAT_1:def_1;
then  not min* Ne in Ke by A1, XBOOLE_0:def_4;
then A3: min* Ne in Ne \ Ke by A2, XBOOLE_0:def_5;
then A4: min* (Ne \ Ke) <= min* Ne by NAT_1:def_1;
A5: Ne \ Ke c= Ne by XBOOLE_1:36;
 min* (Ne \ Ke) in Ne \ Ke by A3, NAT_1:def_1;
then  min* Ne <= min* (Ne \ Ke) by A5, NAT_1:def_1;
hence  min* Ne = min* (Ne \ Ke) by A4, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
hence  min* Ne = min* (Ne \ Ke) ; ::_thesis: verum
end;

theorem Th5: :: STIRL2_1:5
for n being Element of NAT holds
( min* {n} = n & min {n} = n )
proof
let n be Element of NAT ; ::_thesis: ( min* {n} = n & min {n} = n )
A1: min* {n} in {n} by NAT_1:def_1;
 min* {n} = min {n} by Th1;
hence  ( min* {n} = n & min {n} = n ) by A1, TARSKI:def_1; ::_thesis: verum
end;

theorem Th6: :: STIRL2_1:6
for n, k being Element of NAT holds
( min* {n,k} = min (n,k) & min {n,k} = min (n,k) )
proof
let n, k be Element of NAT ; ::_thesis: ( min* {n,k} = min (n,k) & min {n,k} = min (n,k) )
A1: min {n} = n by Th5;
 {n,k} = {n} \/ {k} by ENUMSET1:1;
then A2: min {n,k} = min ((min {n}),(min {k})) by Th2;
 min {n,k} = min* {n,k} by Th1;
hence  ( min* {n,k} = min (n,k) & min {n,k} = min (n,k) ) by A2, A1, Th5; ::_thesis: verum
end;

theorem :: STIRL2_1:7
for n, k, l being Element of NAT holds min* {n,k,l} = min (n,(min (k,l)))
proof
let n, k, l be Element of NAT ; ::_thesis: min* {n,k,l} = min (n,(min (k,l)))
A1: min {n,k,l} = min* {n,k,l} by Th1;
 {n,k,l} = {n} \/ {k,l} by ENUMSET1:2;
then A2: min {n,k,l} = min ((min {n}),(min {k,l})) by Th2;
 min {k,l} = min (k,l) by Th6;
hence  min* {n,k,l} = min (n,(min (k,l))) by A2, A1, Th5; ::_thesis: verum
end;

theorem Th8: :: STIRL2_1:8
for n being Nat holds n is Subset of NAT
proof
let n be Nat; ::_thesis: n is Subset of NAT
now__::_thesis:_for_x_being_set_st_x_in__{__l_where_l_is_Element_of_NAT_:_l_<_n__}__holds_
x_in_NAT
let x be set ; ::_thesis: ( x in  {  l where l is Element of NAT : l < n  }  implies x in NAT )
assume  x in  {  l where l is Element of NAT : l < n  }  ; ::_thesis: x in NAT
then  ex l being Element of NAT st
( x = l & l < n ) ;
hence  x in NAT ; ::_thesis: verum
end;
then   {  l where l is Element of NAT : l < n  }  c= NAT by TARSKI:def_3;
hence  n is Subset of NAT by AXIOMS:4; ::_thesis: verum
end;

registration
let n be Nat;
cluster  ->  natural for Element of n;
coherence
 for b1 being Element of n holds b1 is natural ;
end;

theorem :: STIRL2_1:9
for n being Nat
for N being non empty Subset of NAT st N c= n holds
n - 1 is Element of NAT
proof
let n be Nat; ::_thesis: for N being non empty Subset of NAT st N c= n holds
n - 1 is Element of NAT
let N be non empty Subset of NAT; ::_thesis: ( N c= n implies n - 1 is Element of NAT )
A1: min* N in N by NAT_1:def_1;
assume  N c= n ; ::_thesis: n - 1 is Element of NAT
then  min* N in n by A1;
then  min* N in  {  l where l is Element of NAT : l < n  }  by AXIOMS:4;
then  ex l being Element of NAT st
( min* N = l & l < n ) ;
hence  n - 1 is Element of NAT by NAT_1:20; ::_thesis: verum
end;

theorem Th10: :: STIRL2_1:10
for k, n being Nat st k in n holds
( k <= n - 1 & n - 1 is Element of NAT )
proof
let k, n be Nat; ::_thesis: ( k in n implies ( k <= n - 1 & n - 1 is Element of NAT ) )
assume  k in n ; ::_thesis: ( k <= n - 1 & n - 1 is Element of NAT )
then  k in  {  l where l is Element of NAT : l < n  }  by AXIOMS:4;
then A1: ex l being Element of NAT st
( k = l & l < n ) ;
then A2: n - 1 is Element of NAT by NAT_1:20;
 k < (n - 1) + 1 by A1;
hence  ( k <= n - 1 & n - 1 is Element of NAT ) by A2, NAT_1:13; ::_thesis: verum
end;

theorem :: STIRL2_1:11
for n being Nat holds min* n = 0
proof
let n be Nat; ::_thesis: min* n = 0
now__::_thesis:_min*_n_=_0
percases ( 0 = n or 0 < n ) ;
suppose 0 = n ; ::_thesis: min* n = 0
hence  min* n = 0 by NAT_1:def_1; ::_thesis: verum
end;
suppose 0 < n ; ::_thesis: min* n = 0
then  0 in  {  l where l is Element of NAT : l < n  }  ;
then A1: 0 in n by AXIOMS:4;
 n is Subset of NAT by Th8;
hence  min* n = 0 by A1, NAT_1:def_1; ::_thesis: verum
end;
end;
end;
hence  min* n = 0 ; ::_thesis: verum
end;

theorem Th12: :: STIRL2_1:12
for n being Nat
for N being non empty Subset of NAT st N c= n holds
min* N <= n - 1
proof
let n be Nat; ::_thesis: for N being non empty Subset of NAT st N c= n holds
min* N <= n - 1
let N be non empty Subset of NAT; ::_thesis: ( N c= n implies min* N <= n - 1 )
A1: min* N in N by NAT_1:def_1;
assume  N c= n ; ::_thesis: min* N <= n - 1
hence  min* N <= n - 1 by A1, Th10; ::_thesis: verum
end;

theorem :: STIRL2_1:13
for n being Nat
for N being non empty Subset of NAT st N c= n & N <> {(n - 1)} holds
min* N < n - 1
proof
let n be Nat; ::_thesis: for N being non empty Subset of NAT st N c= n & N <> {(n - 1)} holds
min* N < n - 1
let N be non empty Subset of NAT; ::_thesis: ( N c= n & N <> {(n - 1)} implies min* N < n - 1 )
assume that
A1: N c= n and
A2: N <> {(n - 1)} and
A3: min* N >= n - 1 ; ::_thesis: contradiction
now__::_thesis:_for_k_being_set_st_k_in_N_holds_
k_in_{(n_-_1)}
let k be set ; ::_thesis: ( k in N implies k in {(n - 1)} )
assume A4: k in N ; ::_thesis: k in {(n - 1)}
reconsider k9 = k as Element of NAT by A4;
 min* N <= k9 by A4, NAT_1:def_1;
then A5: n - 1 <= k9 by A3, XXREAL_0:2;
 k9 <= n - 1 by A1, A4, Th10;
then  n - 1 = k by A5, XXREAL_0:1;
hence  k in {(n - 1)} by TARSKI:def_1; ::_thesis: verum
end;
then  N c= {(n - 1)} by TARSKI:def_3;
hence  contradiction by A2, ZFMISC_1:33; ::_thesis: verum
end;

theorem Th14: :: STIRL2_1:14
for n being Nat
for Ne being Subset of NAT st Ne c= n & n > 0 holds
min* Ne <= n - 1
proof
let n be Nat; ::_thesis: for Ne being Subset of NAT st Ne c= n & n > 0 holds
min* Ne <= n - 1
let Ne be Subset of NAT; ::_thesis: ( Ne c= n & n > 0 implies min* Ne <= n - 1 )
assume that
A1: Ne c= n and
A2: n > 0 ; ::_thesis: min* Ne <= n - 1
now__::_thesis:_min*_Ne_<=_n_-_1
percases ( not Ne is empty or Ne is empty ) ;
suppose not Ne is empty ; ::_thesis: min* Ne <= n - 1
hence  min* Ne <= n - 1 by A1, Th12; ::_thesis: verum
end;
suppose Ne is empty ; ::_thesis: min* Ne <= n - 1
then  min* Ne = 0 by NAT_1:def_1;
hence  min* Ne <= n - 1 by A2, NAT_1:20; ::_thesis: verum
end;
end;
end;
hence  min* Ne <= n - 1 ; ::_thesis: verum
end;

definition
let n be Nat;
let X be set ;
let f be Function of n,X;
let x be set ;
:: original: "
redefine funcf " x ->  Subset of NAT;
coherence
 f " x is Subset of NAT
proof
now__::_thesis:_for_y_being_set_st_y_in_f_"_x_holds_
y_in_NAT
A1: n is Subset of NAT by Th8;
let y be set ; ::_thesis: ( y in f " x implies y in NAT )
assume  y in f " x ; ::_thesis: y in NAT
hence  y in NAT by A1, TARSKI:def_3; ::_thesis: verum
end;
hence  f " x is Subset of NAT by TARSKI:def_3; ::_thesis: verum
end;
end;

definition
let X be set ;
let k be Nat;
let f be Function of X,k;
let x be set ;
:: original: .
redefine funcf . x ->  Element of k;
coherence
 f . x is Element of k
proof
percases ( x in dom f or not x in dom f ) ;
suppose x in dom f ; ::_thesis: f . x is Element of k
then  f . x in rng f by FUNCT_1:3;
hence  f . x is Element of k ; ::_thesis: verum
end;
supposeA1: not x in dom f ; ::_thesis: f . x is Element of k
A2: ( k = 0 or k > 0 ) ;
 f . x = 0 by A1, FUNCT_1:def_2;
hence  f . x is Element of k by A2, NAT_1:44, SUBSET_1:def_1; ::_thesis: verum
end;
end;
end;
end;

definition
let n, k be Nat;
let f be Function of n,k;
attrf is "increasing  means :Def1: :: STIRL2_1:def 1
( ( n = 0 implies k = 0 ) & ( k = 0 implies n = 0 ) & ( for l, m being Nat st l in rng f & m in rng f & l < m holds
min* (f " {l}) < min* (f " {m}) ) );
end;

:: deftheorem Def1 defines "increasing STIRL2_1:def_1_:_
 for n, k being Nat
for f being Function of n,k holds
( f is "increasing iff ( ( n = 0 implies k = 0 ) & ( k = 0 implies n = 0 ) & ( for l, m being Nat st l in rng f & m in rng f & l < m holds
min* (f " {l}) < min* (f " {m}) ) ) );

theorem Th15: :: STIRL2_1:15
for n, k being Nat
for f being Function of n,k st n = 0 & k = 0 holds
( f is onto & f is "increasing )
proof
let n, k be Nat; ::_thesis: for f being Function of n,k st n = 0 & k = 0 holds
( f is onto & f is "increasing )
let f be Function of n,k; ::_thesis: ( n = 0 & k = 0 implies ( f is onto & f is "increasing ) )
assume that
A1: n = 0 and
A2: k = 0 ; ::_thesis: ( f is onto & f is "increasing )
A3: for l, m being Nat st l in k & m in k & l < m holds
min* (f " {l}) < min* (f " {m}) by A2;
 rng f = {} by A1;
hence  ( f is onto & f is "increasing ) by A1, A2, A3, Def1, FUNCT_2:def_3; ::_thesis: verum
end;

theorem Th16: :: STIRL2_1:16
for k, n, m being Nat
for f being Function of n,k st n > 0 holds
min* (f " {m}) <= n - 1
proof
let k, n, m be Nat; ::_thesis: for f being Function of n,k st n > 0 holds
min* (f " {m}) <= n - 1
let f be Function of n,k; ::_thesis: ( n > 0 implies min* (f " {m}) <= n - 1 )
A1: f " {m} c= n
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in f " {m} or x in n )
assume  x in f " {m} ; ::_thesis: x in n
then  x in dom f by FUNCT_1:def_7;
hence  x in n ; ::_thesis: verum
end;
assume  n > 0 ; ::_thesis: min* (f " {m}) <= n - 1
hence  min* (f " {m}) <= n - 1 by A1, Th14; ::_thesis: verum
end;

theorem Th17: :: STIRL2_1:17
for n, k being Nat
for f being Function of n,k st f is onto holds
n >= k
proof
let n, k be Nat; ::_thesis: for f being Function of n,k st f is onto holds
n >= k
let f be Function of n,k; ::_thesis: ( f is onto implies n >= k )
assume A1: f is onto ; ::_thesis: n >= k
now__::_thesis:_n_>=_k
percases ( k = 0 or k <> 0 ) ;
suppose k = 0 ; ::_thesis: n >= k
hence  n >= k ; ::_thesis: verum
end;
supposeA2: k <> 0 ; ::_thesis: n >= k
A3: rng f = k by A1, FUNCT_2:def_3;
 dom f = n by A2, FUNCT_2:def_1;
then  card k c= card n by A3, CARD_1:12;
then A4: card k <= card n by NAT_1:39;
 card n = n by CARD_1:def_2;
hence  n >= k by A4, CARD_1:def_2; ::_thesis: verum
end;
end;
end;
hence  n >= k ; ::_thesis: verum
end;

theorem Th18: :: STIRL2_1:18
for n, k being Nat
for f being Function of n,k st f is onto & f is "increasing holds
for m being Nat st m < k holds
m <= min* (f " {m})
proof
let n, k be Nat; ::_thesis: for f being Function of n,k st f is onto & f is "increasing holds
for m being Nat st m < k holds
m <= min* (f " {m})
let f be Function of n,k; ::_thesis: ( f is onto & f is "increasing implies for m being Nat st m < k holds
m <= min* (f " {m}) )
defpred S1[ Nat] means ( $1 < k implies $1 <= min* (f " {$1}) );
assume A1: ( f is onto & f is "increasing ) ; ::_thesis: for m being Nat st m < k holds
m <= min* (f " {m})
A2: for m being Nat st S1[m] holds
S1[m + 1]
proof
A3: k = rng f by A1, FUNCT_2:def_3;
let m be Nat; ::_thesis: ( S1[m] implies S1[m + 1] )
assume A4: S1[m] ; ::_thesis: S1[m + 1]
assume A5: m + 1 < k ; ::_thesis: m + 1 <= min* (f " {(m + 1)})
 m < m + 1 by NAT_1:19;
then  m < k by A5, XXREAL_0:2;
then A6: m in rng f by A3, NAT_1:44;
A7: m < m + 1 by NAT_1:19;
 m + 1 in rng f by A5, A3, NAT_1:44;
then  min* (f " {m}) < min* (f " {(m + 1)}) by A1, A6, A7, Def1;
then  m < min* (f " {(m + 1)}) by A4, A5, A7, XXREAL_0:2;
hence  m + 1 <= min* (f " {(m + 1)}) by NAT_1:13; ::_thesis: verum
end;
A8: S1[ 0 ] ;
 for m being Nat holds S1[m] from NAT_1:sch_2(A8, A2);
hence  for m being Nat st m < k holds
m <= min* (f " {m}) ; ::_thesis: verum
end;

theorem Th19: :: STIRL2_1:19
for k, n being Nat
for f being Function of n,k st f is onto & f is "increasing holds
for m being Nat st m < k holds
min* (f " {m}) <= (n - k) + m
proof
let k, n be Nat; ::_thesis: for f being Function of n,k st f is onto & f is "increasing holds
for m being Nat st m < k holds
min* (f " {m}) <= (n - k) + m
let f be Function of n,k; ::_thesis: ( f is onto & f is "increasing implies for m being Nat st m < k holds
min* (f " {m}) <= (n - k) + m )
assume A1: ( f is onto & f is "increasing ) ; ::_thesis: for m being Nat st m < k holds
min* (f " {m}) <= (n - k) + m
now__::_thesis:_for_m_being_Nat_st_m_<_k_holds_
min*_(f_"_{m})_<=_(n_-_k)_+_m
percases ( k = 0 or k > 0 ) ;
suppose k = 0 ; ::_thesis: for m being Nat st m < k holds
min* (f " {m}) <= (n - k) + m
hence  for m being Nat st m < k holds
min* (f " {m}) <= (n - k) + m ; ::_thesis: verum
end;
suppose k > 0 ; ::_thesis: for m being Nat st m < k holds
min* (f " {m}) <= (n - k) + m
then reconsider k1 = k - 1 as Element of NAT by NAT_1:20;
defpred S1[ Integer] means ( 0 <= $1 implies min* (f " {$1}) <= (n - k) + $1 );
A2: k1 < k1 + 1 by NAT_1:13;
A3: for m being Integer st m <= k1 & S1[m] holds
S1[m - 1]
proof
reconsider nk = n - k as Element of NAT by A1, Th17, NAT_1:21;
A4: k = rng f by A1, FUNCT_2:def_3;
let m be Integer; ::_thesis: ( m <= k1 & S1[m] implies S1[m - 1] )
assume  m <= k1 ; ::_thesis: ( not S1[m] or S1[m - 1] )
then A5: m < k by A2, XXREAL_0:2;
assume A6: S1[m] ; ::_thesis: S1[m - 1]
assume  0 <= m - 1 ; ::_thesis: min* (f " {(m - 1)}) <= (n - k) + (m - 1)
then reconsider m1 = m - 1 as Element of NAT by INT_1:3;
A7: m1 < m1 + 1 by NAT_1:19;
then  m1 < k by A5, XXREAL_0:2;
then A8: m1 in k by NAT_1:44;
 m1 + 1 in k by A5, NAT_1:44;
then  min* (f " {m1}) < min* (f " {(m1 + 1)}) by A1, A7, A8, A4, Def1;
then  min* (f " {m1}) < (nk + m1) + 1 by A6, XXREAL_0:2;
hence  min* (f " {(m - 1)}) <= (n - k) + (m - 1) by NAT_1:13; ::_thesis: verum
end;
A9: S1[k1]
proof
assume  0 <= k1 ; ::_thesis: min* (f " {k1}) <= (n - k) + k1
 ( k = 0 iff n = 0 ) by A1, Def1;
then  min* (f " {k1}) <= n - 1 by Th16;
hence  min* (f " {k1}) <= (n - k) + k1 ; ::_thesis: verum
end;
A10: for m being Integer st m <= k1 holds
S1[m] from INT_1:sch_3(A9, A3);
now__::_thesis:_for_m_being_Nat_st_m_<_k_holds_
min*_(f_"_{m})_<=_(n_-_k)_+_m
let m be Nat; ::_thesis: ( m < k implies min* (f " {m}) <= (n - k) + m )
assume  m < k ; ::_thesis: min* (f " {m}) <= (n - k) + m
then  m < k1 + 1 ;
then  m <= k1 by NAT_1:13;
hence  min* (f " {m}) <= (n - k) + m by A10; ::_thesis: verum
end;
hence  for m being Nat st m < k holds
min* (f " {m}) <= (n - k) + m ; ::_thesis: verum
end;
end;
end;
hence  for m being Nat st m < k holds
min* (f " {m}) <= (n - k) + m ; ::_thesis: verum
end;

theorem Th20: :: STIRL2_1:20
for n, k being Nat
for f being Function of n,k st f is onto & f is "increasing & n = k holds
f = id n
proof
let n, k be Nat; ::_thesis: for f being Function of n,k st f is onto & f is "increasing & n = k holds
f = id n
let f be Function of n,k; ::_thesis: ( f is onto & f is "increasing & n = k implies f = id n )
assume that
A1: ( f is onto & f is "increasing ) and
A2: n = k ; ::_thesis: f = id n
now__::_thesis:_f_=_id_n
percases ( n = 0 or n > 0 ) ;
supposeA3: n = 0 ; ::_thesis: f = id n
A4: for x being set st x in 0 holds
f . x = x ;
 dom f = 0 by A3;
hence  f = id n by A3, A4, FUNCT_1:17; ::_thesis: verum
end;
supposeA5: n > 0 ; ::_thesis: f = id n
A6: now__::_thesis:_for_m9_being_set_st_m9_in_n_holds_
f_._m9_=_m9
let m9 be set ; ::_thesis: ( m9 in n implies f . m9 = m9 )
assume A7: m9 in n ; ::_thesis: f . m9 = m9
 n is Subset of NAT by Th8;
then reconsider m = m9 as Element of NAT by A7;
 m in rng f by A1, A2, A7, FUNCT_2:def_3;
then A8: ex x being set st
( x in dom f & f . x = m ) by FUNCT_1:def_3;
 m in {m} by TARSKI:def_1;
then reconsider F = f " {m} as non empty Subset of NAT by A8, FUNCT_1:def_7;
A9: m < k by A2, A7, NAT_1:44;
then A10: m <= min* (f " {m}) by A1, Th18;
 (n - k) + m = m by A2;
then  min* (f " {m}) <= m by A1, A9, Th19;
then A11: min* F = m by A10, XXREAL_0:1;
 min* F in F by NAT_1:def_1;
then  f . m in {m} by A11, FUNCT_1:def_7;
hence  f . m9 = m9 by TARSKI:def_1; ::_thesis: verum
end;
 dom f = n by A2, A5, FUNCT_2:def_1;
hence  f = id n by A6, FUNCT_1:17; ::_thesis: verum
end;
end;
end;
hence  f = id n ; ::_thesis: verum
end;

theorem :: STIRL2_1:21
for k, n being Nat
for f being Function of n,k st f = id n & n > 0 holds
f is "increasing
proof
let k, n be Nat; ::_thesis: for f being Function of n,k st f = id n & n > 0 holds
f is "increasing
let f be Function of n,k; ::_thesis: ( f = id n & n > 0 implies f is "increasing )
assume that
A1: f = id n and
A2: n > 0 ; ::_thesis: f is "increasing
A3: ex x being set st x in n by A2, XBOOLE_0:def_1;
A4: now__::_thesis:_for_l,_m_being_Nat_st_l_in_rng_f_&_m_in_rng_f_&_l_<_m_holds_
min*_(f_"_{l})_<_min*_(f_"_{m})
let l, m be Nat; ::_thesis: ( l in rng f & m in rng f & l < m implies min* (f " {l}) < min* (f " {m}) )
assume that
A5: l in rng f and
A6: m in rng f and
A7: l < m ; ::_thesis: min* (f " {l}) < min* (f " {m})
A8: ex x being set st f " {l} = {x} by A1, A5, FUNCT_1:74;
A9: l in {l} by TARSKI:def_1;
consider l9 being set  such that
A10: l9 in dom f and
A11: f . l9 = l by A5, FUNCT_1:def_3;
 l = l9 by A1, A10, A11, FUNCT_1:18;
then  l in f " {l} by A10, A11, A9, FUNCT_1:def_7;
then  ( f " {l} = {l} & l in NAT ) by A8, TARSKI:def_1;
then A12: min* (f " {l}) = l by Th5;
A13: m in {m} by TARSKI:def_1;
A14: ex x being set st f " {m} = {x} by A1, A6, FUNCT_1:74;
consider m9 being set  such that
A15: m9 in dom f and
A16: f . m9 = m by A6, FUNCT_1:def_3;
 m = m9 by A1, A15, A16, FUNCT_1:18;
then  m in f " {m} by A15, A16, A13, FUNCT_1:def_7;
then  ( f " {m} = {m} & m in NAT ) by A14, TARSKI:def_1;
hence  min* (f " {l}) < min* (f " {m}) by A7, A12, Th5; ::_thesis: verum
end;
 rng f = n by A1, RELAT_1:45;
hence  f is "increasing by A3, A4, Def1; ::_thesis: verum
end;

theorem :: STIRL2_1:22
for n, k being Nat holds
not ( ( n = 0 implies k = 0 ) & ( k = 0 implies n = 0 ) & ( for f being Function of n,k holds not f is "increasing ) )
proof
let n, k be Nat; ::_thesis: not ( ( n = 0 implies k = 0 ) & ( k = 0 implies n = 0 ) & ( for f being Function of n,k holds not f is "increasing ) )
assume A1: ( n = 0 iff k = 0 ) ; ::_thesis: ex f being Function of n,k st f is "increasing
now__::_thesis:_ex_f_being_Function_of_n,k_st_f_is_"increasing
percases ( n = 0 or n > 0 ) ;
supposeA2: n = 0 ; ::_thesis: ex f being Function of n,k st f is "increasing
set f = {} ;
A3: dom {} = n by A2;
 rng {} = k by A1, A2;
then reconsider f = {} as Function of n,k by A3, FUNCT_2:1;
 f is "increasing by A1, Th15;
hence  ex f being Function of n,k st f is "increasing ; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: ex f being Function of n,k st f is "increasing
then consider f being Function such that
A4: dom f = n and
A5: rng f = {0} by FUNCT_1:5;
reconsider f = f as Function of n,{0} by A4, A5, FUNCT_2:1;
 rng f c= k
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng f or x in k )
assume A6: x in rng f ; ::_thesis: x in k
 x = 0 by A6, TARSKI:def_1;
hence  x in k by A1, A6, NAT_1:44; ::_thesis: verum
end;
then reconsider f = f as Function of n,k by FUNCT_2:6;
 for l, m being Nat st l in rng f & m in rng f & l < m holds
min* (f " {l}) < min* (f " {m}) by A5, TARSKI:def_1;
then  f is "increasing by A1, Def1;
hence  ex f being Function of n,k st f is "increasing ; ::_thesis: verum
end;
end;
end;
hence  ex f being Function of n,k st f is "increasing ; ::_thesis: verum
end;

theorem Th23: :: STIRL2_1:23
for n, k being Nat holds
not ( ( n = 0 implies k = 0 ) & ( k = 0 implies n = 0 ) & n >= k & ( for f being Function of n,k holds
( not f is onto or not f is "increasing ) ) )
proof
let n, k be Nat; ::_thesis: not ( ( n = 0 implies k = 0 ) & ( k = 0 implies n = 0 ) & n >= k & ( for f being Function of n,k holds
( not f is onto or not f is "increasing ) ) )
assume that
A1: ( n = 0 iff k = 0 ) and
A2: n >= k ; ::_thesis: ex f being Function of n,k st
( f is onto & f is "increasing )
now__::_thesis:_ex_f_being_Function_of_n,k_st_
(_f_is_onto_&_f_is_"increasing_)
percases ( n = 0 or n > 0 ) ;
supposeA3: n = 0 ; ::_thesis: ex f being Function of n,k st
( f is onto & f is "increasing )
set f = {} ;
A4: dom {} = n by A3;
 rng {} = k by A1, A3;
then reconsider f = {} as Function of n,k by A4, FUNCT_2:1;
 ( f is onto & f is "increasing ) by A1, Th15;
hence  ex f being Function of n,k st
( f is onto & f is "increasing ) ; ::_thesis: verum
end;
supposeA5: n > 0 ; ::_thesis: ex f being Function of n,k st
( f is onto & f is "increasing )
then reconsider k1 = k - 1 as Element of NAT by A1, NAT_1:20;
reconsider k = k, n = n as non empty Element of NAT by A1, A5, ORDINAL1:def_12;
defpred S1[ Element of n, Element of k] means $2 = min ($1,k1);
A6: for x being Element of n ex y being Element of k st S1[x,y]
proof
let x be Element of n; ::_thesis: ex y being Element of k st S1[x,y]
A7: k1 < k1 + 1 by NAT_1:13;
 min (x,k1) <= k1 by XXREAL_0:17;
then  min (x,k1) < k by A7, XXREAL_0:2;
then  min (x,k1) in k by NAT_1:44;
hence  ex y being Element of k st S1[x,y] ; ::_thesis: verum
end;
consider f being Function of n,k such that
A8: for m being Element of n holds S1[m,f . m] from FUNCT_2:sch_3(A6);
now__::_thesis:_for_m9_being_set_st_m9_in_k_holds_
m9_in_rng_f
let m9 be set ; ::_thesis: ( m9 in k implies m9 in rng f )
assume A9: m9 in k ; ::_thesis: m9 in rng f
 k is Subset of NAT by Th8;
then reconsider m = m9 as Element of NAT by A9;
A10: m < k1 + 1 by A9, NAT_1:44;
then  m < n by A2, XXREAL_0:2;
then A11: m in n by NAT_1:44;
then A12: m in dom f by FUNCT_2:def_1;
 m <= k1 by A10, NAT_1:13;
then  min (m,k1) = m by XXREAL_0:def_9;
then  f . m = m by A8, A11;
hence  m9 in rng f by A12, FUNCT_1:def_3; ::_thesis: verum
end;
then  k c= rng f by TARSKI:def_3;
then  k = rng f by XBOOLE_0:def_10;
then A13: f is onto by FUNCT_2:def_3;
A14: for m being Nat st m in rng f holds
min* (f " {m}) = m
proof
let m be Nat; ::_thesis: ( m in rng f implies min* (f " {m}) = m )
assume  m in rng f ; ::_thesis: min* (f " {m}) = m
then A15: m < k1 + 1 by NAT_1:44;
then A16: m <= k1 by NAT_1:13;
 m < n by A2, A15, XXREAL_0:2;
then A17: m in n by NAT_1:44;
then A18: m in dom f by FUNCT_2:def_1;
 m <= k1 by A15, NAT_1:13;
then  min (m,k1) = m by XXREAL_0:def_9;
then  f . m = m by A8, A17;
then A19: f . m in {m} by TARSKI:def_1;
then A20: m in f " {m} by A18, FUNCT_1:def_7;
A21: not f " {m} is empty by A19, A18, FUNCT_1:def_7;
now__::_thesis:_min*_(f_"_{m})_=_m
percases ( m < k1 or m = k1 ) by A16, XXREAL_0:1;
supposeA22: m < k1 ; ::_thesis: min* (f " {m}) = m
now__::_thesis:_for_l9_being_set_st_l9_in_f_"_{m}_holds_
l9_in_{m}
A23: n is Subset of NAT by Th8;
let l9 be set ; ::_thesis: ( l9 in f " {m} implies l9 in {m} )
assume A24: l9 in f " {m} ; ::_thesis: l9 in {m}
 l9 in dom f by A24, FUNCT_1:def_7;
then  l9 in n ;
then reconsider l = l9 as Element of NAT by A23;
 f . l in {m} by A24, FUNCT_1:def_7;
then A25: f . l = m by TARSKI:def_1;
 l in dom f by A24, FUNCT_1:def_7;
then  min (l,k1) = m by A8, A25;
then  l = m by A22, XXREAL_0:15;
hence  l9 in {m} by TARSKI:def_1; ::_thesis: verum
end;
then A26: f " {m} c= {m} by TARSKI:def_3;
 min* (f " {m}) in f " {m} by A21, NAT_1:def_1;
hence  min* (f " {m}) = m by A26, TARSKI:def_1; ::_thesis: verum
end;
supposeA27: m = k1 ; ::_thesis: min* (f " {m}) = m
 for l being Nat st l in f " {m} holds
m <= l
proof
let l be Nat; ::_thesis: ( l in f " {m} implies m <= l )
assume A28: l in f " {m} ; ::_thesis: m <= l
 f . l in {m} by A28, FUNCT_1:def_7;
then A29: f . l = m by TARSKI:def_1;
 l in dom f by A28, FUNCT_1:def_7;
then  f . l = min (l,m) by A8, A27;
hence  m <= l by A29, XXREAL_0:def_9; ::_thesis: verum
end;
hence  min* (f " {m}) = m by A20, NAT_1:def_1; ::_thesis: verum
end;
end;
end;
hence  min* (f " {m}) = m ; ::_thesis: verum
end;
now__::_thesis:_for_l,_m_being_Nat_st_l_in_rng_f_&_m_in_rng_f_&_l_<_m_holds_
min*_(f_"_{l})_<_min*_(f_"_{m})
let l, m be Nat; ::_thesis: ( l in rng f & m in rng f & l < m implies min* (f " {l}) < min* (f " {m}) )
assume that
A30: l in rng f and
A31: m in rng f and
A32: l < m ; ::_thesis: min* (f " {l}) < min* (f " {m})
 min* (f " {l}) = l by A14, A30;
hence  min* (f " {l}) < min* (f " {m}) by A14, A31, A32; ::_thesis: verum
end;
then  f is "increasing by Def1;
hence  ex f being Function of n,k st
( f is onto & f is "increasing ) by A13; ::_thesis: verum
end;
end;
end;
hence  ex f being Function of n,k st
( f is onto & f is "increasing ) ; ::_thesis: verum
end;

scheme :: STIRL2_1:sch 1
Sch1{ F1() ->  Nat, F2() ->  Nat, P1[ set ] } :
  {  f where f is Function of F1(),F2() : P1[f]  }  is finite
proof
set F =  {  f where f is Function of F1(),F2() : P1[f]  }  ;
now__::_thesis:__{__f_where_f_is_Function_of_F1(),F2()_:_P1[f]__}__is_finite
percases ( F2() = 0 or F2() > 0 ) ;
supposeA1: F2() = 0 ; ::_thesis:  {  f where f is Function of F1(),F2() : P1[f]  }  is finite
  {  f where f is Function of F1(),F2() : P1[f]  }  c= {{}}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of F1(),F2() : P1[f]  }  or x in {{}} )
assume  x in  {  f where f is Function of F1(),F2() : P1[f]  }  ; ::_thesis: x in {{}}
then consider f being Function of F1(),F2() such that
A2: x = f and
 P1[f] ;
 f = {} by A1;
hence  x in {{}} by A2, TARSKI:def_1; ::_thesis: verum
end;
hence   {  f where f is Function of F1(),F2() : P1[f]  }  is finite ; ::_thesis: verum
end;
supposeA3: F2() > 0 ; ::_thesis:  {  f where f is Function of F1(),F2() : P1[f]  }  is finite
A4:  {  f where f is Function of F1(),F2() : P1[f]  }  c= Funcs (F1(),F2())
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of F1(),F2() : P1[f]  }  or x in Funcs (F1(),F2()) )
assume  x in  {  f where f is Function of F1(),F2() : P1[f]  }  ; ::_thesis: x in Funcs (F1(),F2())
then  ex f being Function of F1(),F2() st
( x = f & P1[f] ) ;
hence  x in Funcs (F1(),F2()) by A3, FUNCT_2:8; ::_thesis: verum
end;
 Funcs (F1(),F2()) is finite by FRAENKEL:6;
hence   {  f where f is Function of F1(),F2() : P1[f]  }  is finite by A4; ::_thesis: verum
end;
end;
end;
hence   {  f where f is Function of F1(),F2() : P1[f]  }  is finite ; ::_thesis: verum
end;

theorem Th24: :: STIRL2_1:24
for n, k being Nat holds  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  is finite
proof
let n, k be Nat; ::_thesis:  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  is finite
defpred S1[ Function of n,k] means ( $1 is onto & $1 is "increasing );
  {  f where f is Function of n,k : S1[f]  }  is finite from STIRL2_1:sch_1();
hence   {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  is finite ; ::_thesis: verum
end;

theorem Th25: :: STIRL2_1:25
for n, k being Nat holds card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  is Element of NAT
proof
let n, k be Nat; ::_thesis: card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  is Element of NAT
set F =  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  ;
consider m being Nat such that
A1:  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  ,m are_equipotent by Th24, CARD_1:43;
 card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  = card m by A1, CARD_1:5;
hence  card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  is Element of NAT ; ::_thesis: verum
end;

definition
let n, k be Nat;
funcn block k ->  Element of NAT  equals :: STIRL2_1:def 2
 card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  ;
coherence
 card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  is Element of NAT by Th25;
end;

:: deftheorem  defines block STIRL2_1:def_2_:_
 for n, k being Nat holds n block k = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  ;

theorem Th26: :: STIRL2_1:26
for n being Nat holds n block n = 1
proof
let n be Nat; ::_thesis: n block n = 1
set F =  {  f where f is Function of n,n : ( f is onto & f is "increasing )  }  ;
A1:  {  f where f is Function of n,n : ( f is onto & f is "increasing )  }  c= {(id n)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of n,n : ( f is onto & f is "increasing )  }  or x in {(id n)} )
assume  x in  {  f where f is Function of n,n : ( f is onto & f is "increasing )  }  ; ::_thesis: x in {(id n)}
then consider f being Function of n,n such that
A2: x = f and
A3: ( f is onto & f is "increasing ) ;
 f = id n by A3, Th20;
hence  x in {(id n)} by A2, TARSKI:def_1; ::_thesis: verum
end;
 ( n = 0 iff n = 0 ) ;
then consider f being Function of n,n such that
A4: ( f is onto & f is "increasing ) by Th23;
 f in  {  f where f is Function of n,n : ( f is onto & f is "increasing )  }  by A4;
then   {  f where f is Function of n,n : ( f is onto & f is "increasing )  }  = {(id n)} by A1, ZFMISC_1:33;
hence  n block n = 1 by CARD_1:30; ::_thesis: verum
end;

theorem Th27: :: STIRL2_1:27
for k being Nat st k <> 0 holds
0 block k = 0
proof
let k be Nat; ::_thesis: ( k <> 0 implies 0 block k = 0 )
set F =  {  f where f is Function of 0,k : ( f is onto & f is "increasing )  }  ;
assume A1: k <> 0 ; ::_thesis: 0 block k = 0
  {  f where f is Function of 0,k : ( f is onto & f is "increasing )  }  = {}
proof
assume   {  f where f is Function of 0,k : ( f is onto & f is "increasing )  }  <> {} ; ::_thesis: contradiction
then consider x being set  such that
A2: x in  {  f where f is Function of 0,k : ( f is onto & f is "increasing )  }  by XBOOLE_0:def_1;
 ex f being Function of 0,k st
( x = f & f is onto & f is "increasing ) by A2;
hence  contradiction by A1, Def1; ::_thesis: verum
end;
hence  0 block k = 0 ; ::_thesis: verum
end;

theorem :: STIRL2_1:28
for k being Nat holds
( 0 block k = 1 iff k = 0 ) by Th26, Th27;

theorem Th29: :: STIRL2_1:29
for n, k being Nat st n < k holds
n block k = 0
proof
let n, k be Nat; ::_thesis: ( n < k implies n block k = 0 )
set F =  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  ;
assume A1: n < k ; ::_thesis: n block k = 0
  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  = {}
proof
assume   {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  <> {} ; ::_thesis: contradiction
then consider x being set  such that
A2: x in  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  by XBOOLE_0:def_1;
 ex f being Function of n,k st
( x = f & f is onto & f is "increasing ) by A2;
hence  contradiction by A1, Th17; ::_thesis: verum
end;
hence  n block k = 0 ; ::_thesis: verum
end;

theorem :: STIRL2_1:30
for n being Nat holds
( n block 0 = 1 iff n = 0 )
proof
let n be Nat; ::_thesis: ( n block 0 = 1 iff n = 0 )
 ( n block 0 = 1 implies n = 0 )
proof
set F =  {  f where f is Function of n,0 : ( f is onto & f is "increasing )  }  ;
A1: card {{}} = 1 by CARD_1:30;
assume  n block 0 = 1 ; ::_thesis: n = 0
then consider x being set  such that
A2:  {  f where f is Function of n,0 : ( f is onto & f is "increasing )  }  = {x} by A1, CARD_1:29;
 x in  {  f where f is Function of n,0 : ( f is onto & f is "increasing )  }  by A2, TARSKI:def_1;
then  ex f being Function of n,0 st
( x = f & f is onto & f is "increasing ) ;
hence  n = 0 by Def1; ::_thesis: verum
end;
hence  ( n block 0 = 1 iff n = 0 ) by Th26; ::_thesis: verum
end;

theorem Th31: :: STIRL2_1:31
for n being Nat st n <> 0 holds
n block 0 = 0
proof
let n be Nat; ::_thesis: ( n <> 0 implies n block 0 = 0 )
set F =  {  f where f is Function of n,0 : ( f is onto & f is "increasing )  }  ;
assume A1: n <> 0 ; ::_thesis: n block 0 = 0
  {  f where f is Function of n,0 : ( f is onto & f is "increasing )  }  = {}
proof
assume   {  f where f is Function of n,0 : ( f is onto & f is "increasing )  }  <> {} ; ::_thesis: contradiction
then consider x being set  such that
A2: x in  {  f where f is Function of n,0 : ( f is onto & f is "increasing )  }  by XBOOLE_0:def_1;
 ex f being Function of n,0 st
( x = f & f is onto & f is "increasing ) by A2;
hence  contradiction by A1, Def1; ::_thesis: verum
end;
hence  n block 0 = 0 ; ::_thesis: verum
end;

theorem Th32: :: STIRL2_1:32
for n being Nat st n <> 0 holds
n block 1 = 1
proof
let n be Nat; ::_thesis: ( n <> 0 implies n block 1 = 1 )
set F =  {  g where g is Function of n,1 : ( g is onto & g is "increasing )  }  ;
assume  n <> 0 ; ::_thesis: n block 1 = 1
then  n >= 1 + 0 by NAT_1:13;
then consider f being Function of n,1 such that
A1: ( f is onto & f is "increasing ) by Th23;
A2:  {  g where g is Function of n,1 : ( g is onto & g is "increasing )  }  c= {f}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  g where g is Function of n,1 : ( g is onto & g is "increasing )  }  or x in {f} )
assume  x in  {  g where g is Function of n,1 : ( g is onto & g is "increasing )  }  ; ::_thesis: x in {f}
then consider g being Function of n,1 such that
A3: x = g and
 ( g is onto & g is "increasing ) ;
 f = g by CARD_1:49, FUNCT_2:51;
hence  x in {f} by A3, TARSKI:def_1; ::_thesis: verum
end;
 f in  {  g where g is Function of n,1 : ( g is onto & g is "increasing )  }  by A1;
then   {  g where g is Function of n,1 : ( g is onto & g is "increasing )  }  = {f} by A2, ZFMISC_1:33;
hence  n block 1 = 1 by CARD_1:30; ::_thesis: verum
end;

theorem :: STIRL2_1:33
for k, n being Nat holds
( ( ( 1 <= k & k <= n ) or k = n ) iff n block k > 0 )
proof
let k, n be Nat; ::_thesis: ( ( ( 1 <= k & k <= n ) or k = n ) iff n block k > 0 )
thus  ( ( ( 1 <= k & k <= n ) or k = n ) implies n block k > 0 ) ::_thesis: ( not n block k > 0 or ( 1 <= k & k <= n ) or k = n )
proof
set F =  {  g where g is Function of n,k : ( g is onto & g is "increasing )  }  ;
assume A1: ( ( 1 <= k & k <= n ) or k = n ) ; ::_thesis: n block k > 0
 ( k = 0 iff n = 0 ) by A1;
then consider f being Function of n,k such that
A2: ( f is onto & f is "increasing ) by A1, Th23;
 f in  {  g where g is Function of n,k : ( g is onto & g is "increasing )  }  by A2;
hence  n block k > 0 ; ::_thesis: verum
end;
thus  ( not n block k > 0 or ( 1 <= k & k <= n ) or k = n ) ::_thesis: verum
proof
assume A3: n block k > 0 ; ::_thesis: ( ( 1 <= k & k <= n ) or k = n )
assume A4: ( not ( 1 <= k & k <= n ) & not k = n ) ; ::_thesis: contradiction
then  ( 1 + 0 > k or k > n ) ;
then  ( ( k = 0 & n <> k ) or k > n ) by A4, NAT_1:13;
hence  contradiction by A3, Th29, Th31; ::_thesis: verum
end;
end;


scheme :: STIRL2_1:sch 2
Sch2{ F1() ->  set , F2() ->  set , F3() ->  set , F4() ->  set , F5() ->  Function of F1(),F2(), F6( set ) ->  set } :
 ex h being Function of F3(),F4() st
( h | F1() = F5() & ( for x being set st x in F3() \ F1() holds
h . x = F6(x) ) )
provided
A1: for x being set st x in F3() \ F1() holds
F6(x) in F4() and
A2: ( F1() c= F3() & F2() c= F4() ) and
A3: ( F2() is empty implies F1() is empty )
proof
defpred S1[ set , set ] means ( ( $1 in F1() implies $2 = F5() . $1 ) & ( $1 in F3() \ F1() implies $2 = F6($1) ) );
A4: for x being set st x in F3() holds
ex y being set st
( y in F4() & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in F3() implies ex y being set st
( y in F4() & S1[x,y] ) )
assume A5: x in F3() ; ::_thesis: ex y being set st
( y in F4() & S1[x,y] )
now__::_thesis:_ex_y_being_set_st_
(_y_in_F4()_&_S1[x,y]_)
percases ( x in F1() or not x in F1() ) ;
supposeA6: x in F1() ; ::_thesis: ex y being set st
( y in F4() & S1[x,y] )
then  x in dom F5() by A3, FUNCT_2:def_1;
then  F5() . x in rng F5() by FUNCT_1:def_3;
then A7: F5() . x in F4() by A2, TARSKI:def_3;
 not x in F3() \ F1() by A6, XBOOLE_0:def_5;
hence  ex y being set st
( y in F4() & S1[x,y] ) by A7; ::_thesis: verum
end;
supposeA8: not x in F1() ; ::_thesis: ex y being set st
( y in F4() & S1[x,y] )
then  x in F3() \ F1() by A5, XBOOLE_0:def_5;
then  F6(x) in F4() by A1;
hence  ex y being set st
( y in F4() & S1[x,y] ) by A8; ::_thesis: verum
end;
end;
end;
hence  ex y being set st
( y in F4() & S1[x,y] ) ; ::_thesis: verum
end;
consider h being Function of F3(),F4() such that
A9: for x being set st x in F3() holds
S1[x,h . x] from FUNCT_2:sch_1(A4);
A10: dom F5() = (dom h) /\ F1()
proof
now__::_thesis:_dom_F5()_=_(dom_h)_/\_F1()
percases ( F2() is empty or not F2() is empty ) ;
suppose F2() is empty ; ::_thesis: dom F5() = (dom h) /\ F1()
hence  dom F5() = (dom h) /\ F1() by A3; ::_thesis: verum
end;
supposeA11: not F2() is empty ; ::_thesis: dom F5() = (dom h) /\ F1()
then  not F4() is empty by A2;
then A12: dom h = F3() by FUNCT_2:def_1;
 dom F5() = F1() by A11, FUNCT_2:def_1;
hence  dom F5() = (dom h) /\ F1() by A2, A12, XBOOLE_1:28; ::_thesis: verum
end;
end;
end;
hence  dom F5() = (dom h) /\ F1() ; ::_thesis: verum
end;
take  h ; ::_thesis: ( h | F1() = F5() & ( for x being set st x in F3() \ F1() holds
h . x = F6(x) ) )
 for x being set st x in dom F5() holds
h . x = F5() . x
proof
let x be set ; ::_thesis: ( x in dom F5() implies h . x = F5() . x )
assume  x in dom F5() ; ::_thesis: h . x = F5() . x
then  x in F1() ;
hence  h . x = F5() . x by A2, A9; ::_thesis: verum
end;
hence  h | F1() = F5() by A10, FUNCT_1:46; ::_thesis: for x being set st x in F3() \ F1() holds
h . x = F6(x)
thus  for x being set st x in F3() \ F1() holds
h . x = F6(x) by A9; ::_thesis: verum
end;

scheme :: STIRL2_1:sch 3
Sch3{ F1() ->  set , F2() ->  set , F3() ->  set , F4() ->  set , F5( set ) ->  set , P1[ set , set , set ] } :
 card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }
provided
A1: for x being set st x in F3() \ F1() holds
F5(x) in F4() and
A2: ( F1() c= F3() & F2() c= F4() ) and
A3: ( F2() is empty implies F1() is empty ) and
A4: for f being Function of F3(),F4() st ( for x being set st x in F3() \ F1() holds
F5(x) = f . x ) holds
( P1[f,F3(),F4()] iff P1[f | F1(),F1(),F2()] )
proof
defpred S1[ set , set ] means  for f being Function of F1(),F2()
for h being Function of F3(),F4() st f = $1 & h = $2 holds
h | F1() = f;
set F2 =  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  ;
set F1 =  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  ;
A5: for f9 being set st f9 in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  holds
ex g9 being set st
( g9 in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  & S1[f9,g9] )
proof
let f9 be set ; ::_thesis: ( f9 in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  implies ex g9 being set st
( g9 in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  & S1[f9,g9] ) )
assume  f9 in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  ; ::_thesis: ex g9 being set st
( g9 in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  & S1[f9,g9] )
then consider f being Function of F1(),F2() such that
A6: f = f9 and
A7: P1[f,F1(),F2()] ;
consider h being Function of F3(),F4() such that
A8: ( h | F1() = f & ( for x being set st x in F3() \ F1() holds
h . x = F5(x) ) ) from STIRL2_1:sch_2(A1, A2, A3);
A9: S1[f9,h] by A6, A8;
A10: rng f c= F2() ;
 P1[h,F3(),F4()] by A4, A7, A8;
then  h in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  by A8, A10;
hence  ex g9 being set st
( g9 in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  & S1[f9,g9] ) by A9; ::_thesis: verum
end;
consider ff being Function of  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  , {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }   such that
A11: for x being set st x in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  holds
S1[x,ff . x] from FUNCT_2:sch_1(A5);
A12: ( F4() is empty implies F3() is empty )
proof
assume A13: F4() is empty ; ::_thesis: F3() is empty
assume  not F3() is empty ; ::_thesis: contradiction
then  ex x being set st x in F3() by XBOOLE_0:def_1;
hence  contradiction by A1, A2, A3, A13; ::_thesis: verum
end;
now__::_thesis:_card__{__f_where_f_is_Function_of_F1(),F2()_:_P1[f,F1(),F2()]__}__=_card__{__f_where_f_is_Function_of_F3(),F4()_:_(_P1[f,F3(),F4()]_&_rng_(f_|_F1())_c=_F2()_&_(_for_x_being_set_st_x_in_F3()_\_F1()_holds_
f_._x_=_F5(x)_)_)__}_
percases ( F4() is empty or not F4() is empty ) ;
supposeA14: F4() is empty ; ::_thesis: card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }
set Empty = {} ;
A15:  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  c= {{}}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  or x in {{}} )
assume  x in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  ; ::_thesis: x in {{}}
then  ex f being Function of F3(),F4() st
( f = x & P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) ) ;
then  x = {} by A14;
hence  x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
A16:  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  c= {{}}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  or x in {{}} )
assume  x in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  ; ::_thesis: x in {{}}
then  ex f being Function of F1(),F2() st
( f = x & P1[f,F1(),F2()] ) ;
then  x = {} by A2, A14;
hence  x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
now__::_thesis:_card__{__f_where_f_is_Function_of_F1(),F2()_:_P1[f,F1(),F2()]__}__=_card__{__f_where_f_is_Function_of_F3(),F4()_:_(_P1[f,F3(),F4()]_&_rng_(f_|_F1())_c=_F2()_&_(_for_x_being_set_st_x_in_F3()_\_F1()_holds_
f_._x_=_F5(x)_)_)__}_
percases ( P1[ {} , {} , {} ] or not P1[ {} , {} , {} ] ) ;
supposeA17: P1[ {} , {} , {} ] ; ::_thesis: card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }
A18: rng {} = {} ;
A19: F2() is empty by A2, A14;
 dom {} = {} ;
then  {} is Function of F1(),F2() by A3, A19, A18, FUNCT_2:1;
then  {} in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  by A3, A17, A19;
then A20:  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = {{}} by A16, ZFMISC_1:33;
A21: rng {} = {} ;
 dom {} = {} ;
then reconsider Empty = {} as Function of F3(),F4() by A12, A14, A21, FUNCT_2:1;
A22: for x being set st x in F3() \ F1() holds
Empty . x = F5(x) by A12;
 rng (Empty | F1()) c= F2() by A2, A14;
then  {} in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  by A12, A14, A17, A22;
hence  card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  by A15, A20, ZFMISC_1:33; ::_thesis: verum
end;
supposeA23: P1[ {} , {} , {} ] ; ::_thesis: card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }
A24: not {} in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }
proof
assume  {} in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  ; ::_thesis: contradiction
then  ex f being Function of F1(),F2() st
( f = {} & P1[f,F1(),F2()] ) ;
hence  contradiction by A2, A3, A14, A23; ::_thesis: verum
end;
A25: (  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  = {} or  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  = {{}} ) by A15, ZFMISC_1:33;
A26: not {} in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }
proof
assume  {} in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  ; ::_thesis: contradiction
then  ex f being Function of F3(),F4() st
( f = {} & P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) ) ;
hence  contradiction by A12, A14, A23; ::_thesis: verum
end;
 (  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = {} or  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = {{}} ) by A16, ZFMISC_1:33;
hence  card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  by A24, A26, A25, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence  card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  ; ::_thesis: verum
end;
supposeA27: not F4() is empty ; ::_thesis: card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }
now__::_thesis:_card__{__f_where_f_is_Function_of_F1(),F2()_:_P1[f,F1(),F2()]__}__=_card__{__f_where_f_is_Function_of_F3(),F4()_:_(_P1[f,F3(),F4()]_&_rng_(f_|_F1())_c=_F2()_&_(_for_x_being_set_st_x_in_F3()_\_F1()_holds_
f_._x_=_F5(x)_)_)__}_
percases (  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  is empty or not  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  is empty ) ;
supposeA28:  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  is empty ; ::_thesis: card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }
  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  is empty
proof
assume  not  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  is empty ; ::_thesis: contradiction
then consider x being set  such that
A29: x in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  by XBOOLE_0:def_1;
consider f being Function of F1(),F2() such that
 f = x and
A30: P1[f,F1(),F2()] by A29;
A31: rng f c= F2() ;
consider h being Function of F3(),F4() such that
A32: ( h | F1() = f & ( for x being set st x in F3() \ F1() holds
h . x = F5(x) ) ) from STIRL2_1:sch_2(A1, A2, A3);
 P1[h,F3(),F4()] by A4, A30, A32;
then  h in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  by A32, A31;
hence  contradiction by A28; ::_thesis: verum
end;
hence  card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  by A28; ::_thesis: verum
end;
supposeA33: not  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  is empty ; ::_thesis: card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }
  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  c= rng ff
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  or y in rng ff )
assume  y in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  ; ::_thesis: y in rng ff
then consider f being Function of F3(),F4() such that
A34: f = y and
A35: P1[f,F3(),F4()] and
A36: rng (f | F1()) c= F2() and
A37: for x being set st x in F3() \ F1() holds
f . x = F5(x) ;
A38: dom (f | F1()) = (dom f) /\ F1() by RELAT_1:61;
 dom f = F3() by A27, FUNCT_2:def_1;
then A39: dom (f | F1()) = F1() by A2, A38, XBOOLE_1:28;
then reconsider h = f | F1() as Function of F1(),(rng (f | F1())) by FUNCT_2:1;
 ( rng (f | F1()) is empty implies F1() is empty ) by A39, RELAT_1:42;
then reconsider h = h as Function of F1(),F2() by A36, FUNCT_2:6;
 P1[f | F1(),F1(),F2()] by A4, A35, A37;
then A40: h in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  ;
A41: dom ff =  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  by A33, FUNCT_2:def_1;
then  ff . h in rng ff by A40, FUNCT_1:def_3;
then  ff . h in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  ;
then consider ffh being Function of F3(),F4() such that
A42: ffh = ff . h and
 P1[ffh,F3(),F4()] and
 rng (ffh | F1()) c= F2() and
A43: for x being set st x in F3() \ F1() holds
ffh . x = F5(x) ;
now__::_thesis:_for_x_being_set_st_x_in_F3()_holds_
ffh_._x_=_f_._x
let x be set ; ::_thesis: ( x in F3() implies ffh . x = f . x )
assume A44: x in F3() ; ::_thesis: ffh . x = f . x
now__::_thesis:_ffh_._x_=_f_._x
percases ( x in F1() or not x in F1() ) ;
suppose x in F1() ; ::_thesis: ffh . x = f . x
then A45: x in dom h by A3, FUNCT_2:def_1;
 ffh | F1() = h by A11, A40, A42;
then  h . x = ffh . x by A45, FUNCT_1:47;
hence  ffh . x = f . x by A45, FUNCT_1:47; ::_thesis: verum
end;
suppose not x in F1() ; ::_thesis: ffh . x = f . x
then A46: x in F3() \ F1() by A44, XBOOLE_0:def_5;
then  ffh . x = F5(x) by A43;
hence  ffh . x = f . x by A37, A46; ::_thesis: verum
end;
end;
end;
hence  ffh . x = f . x ; ::_thesis: verum
end;
then  ffh = f by FUNCT_2:12;
hence  y in rng ff by A34, A40, A41, A42, FUNCT_1:def_3; ::_thesis: verum
end;
then A47: rng ff =  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  by XBOOLE_0:def_10;
 for x1, x2 being set st x1 in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  & x2 in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  & ff . x1 = ff . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  & x2 in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  & ff . x1 = ff . x2 implies x1 = x2 )
assume that
A48: x1 in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  and
A49: x2 in  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  and
A50: ff . x1 = ff . x2 ; ::_thesis: x1 = x2
A51: ex f2 being Function of F1(),F2() st
( x2 = f2 & P1[f2,F1(),F2()] ) by A49;
 dom ff =  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  by A33, FUNCT_2:def_1;
then  ff . x1 in rng ff by A48, FUNCT_1:def_3;
then  ff . x1 in  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  ;
then consider F1 being Function of F3(),F4() such that
A52: ff . x1 = F1 and
 P1[F1,F3(),F4()] and
 rng (F1 | F1()) c= F2() and
 for x being set st x in F3() \ F1() holds
F1 . x = F5(x) ;
consider f1 being Function of F1(),F2() such that
A53: x1 = f1 and
 P1[f1,F1(),F2()] by A48;
 F1 | F1() = f1 by A11, A48, A53, A52;
hence  x1 = x2 by A11, A49, A50, A53, A51, A52; ::_thesis: verum
end;
then A54: ff is one-to-one by A33, FUNCT_2:19;
 dom ff =  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  by A33, FUNCT_2:def_1;
then   {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  , {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  are_equipotent by A47, A54, WELLORD2:def_4;
hence  card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  by CARD_1:5; ::_thesis: verum
end;
end;
end;
hence  card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  ; ::_thesis: verum
end;
end;
end;
hence  card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of F3(),F4() : ( P1[f,F3(),F4()] & rng (f | F1()) c= F2() & ( for x being set st x in F3() \ F1() holds
f . x = F5(x) ) )  }  ; ::_thesis: verum
end;

scheme :: STIRL2_1:sch 4
Sch4{ F1() ->  set , F2() ->  set , F3() ->  set , F4() ->  set , P1[ set , set , set ] } :
 card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & f . F3() = F4() )  }
provided
A1: ( F2() is empty implies F1() is empty ) and
A2: not F3() in F1() and
A3: for f being Function of (F1() \/ {F3()}),(F2() \/ {F4()}) st f . F3() = F4() holds
( P1[f,F1() \/ {F3()},F2() \/ {F4()}] iff P1[f | F1(),F1(),F2()] )
proof
set Y1 = F2() \/ {F4()};
set X1 = F1() \/ {F3()};
deffunc H1( set ) ->  set = F4();
A4: for f being Function of (F1() \/ {F3()}),(F2() \/ {F4()}) holds
( ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
H1(x) = f . x ) iff f . F3() = F4() )
proof
let f be Function of (F1() \/ {F3()}),(F2() \/ {F4()}); ::_thesis: ( ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
H1(x) = f . x ) iff f . F3() = F4() )
A5: (F1() \/ {F3()}) \ F1() = {F3()} \ F1() by XBOOLE_1:40
.= {F3()} by A2, ZFMISC_1:59 ;
thus  ( ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
H1(x) = f . x ) implies f . F3() = F4() ) ::_thesis: ( f . F3() = F4() implies for x being set st x in (F1() \/ {F3()}) \ F1() holds
H1(x) = f . x )
proof
A6: F3() in {F3()} by TARSKI:def_1;
assume  for x being set st x in (F1() \/ {F3()}) \ F1() holds
H1(x) = f . x ; ::_thesis: f . F3() = F4()
hence  f . F3() = F4() by A5, A6; ::_thesis: verum
end;
thus  ( f . F3() = F4() implies for x being set st x in (F1() \/ {F3()}) \ F1() holds
H1(x) = f . x ) by A5, TARSKI:def_1; ::_thesis: verum
end;
A7: for f being Function of (F1() \/ {F3()}),(F2() \/ {F4()}) st ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
H1(x) = f . x ) holds
( P1[f,F1() \/ {F3()},F2() \/ {F4()}] iff P1[f | F1(),F1(),F2()] )
proof
let f be Function of (F1() \/ {F3()}),(F2() \/ {F4()}); ::_thesis: ( ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
H1(x) = f . x ) implies ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] iff P1[f | F1(),F1(),F2()] ) )
assume  for x being set st x in (F1() \/ {F3()}) \ F1() holds
H1(x) = f . x ; ::_thesis: ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] iff P1[f | F1(),F1(),F2()] )
then  F4() = f . F3() by A4;
hence  ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] iff P1[f | F1(),F1(),F2()] ) by A3; ::_thesis: verum
end;
set F2 =  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & f . F3() = F4() )  }  ;
set F1 =  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
f . x = H1(x) ) )  }  ;
A8: for x being set st x in (F1() \/ {F3()}) \ F1() holds
H1(x) in F2() \/ {F4()}
proof
let x be set ; ::_thesis: ( x in (F1() \/ {F3()}) \ F1() implies H1(x) in F2() \/ {F4()} )
assume  x in (F1() \/ {F3()}) \ F1() ; ::_thesis: H1(x) in F2() \/ {F4()}
A9: {F4()} c= F2() \/ {F4()} by XBOOLE_1:7;
 F4() in {F4()} by TARSKI:def_1;
hence  H1(x) in F2() \/ {F4()} by A9; ::_thesis: verum
end;
A10:  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
f . x = H1(x) ) )  }  c=  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & f . F3() = F4() )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
f . x = H1(x) ) )  }  or x in  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & f . F3() = F4() )  }  )
assume  x in  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
f . x = H1(x) ) )  }  ; ::_thesis: x in  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & f . F3() = F4() )  }
then consider f being Function of (F1() \/ {F3()}),(F2() \/ {F4()}) such that
A11: x = f and
A12: P1[f,F1() \/ {F3()},F2() \/ {F4()}] and
A13: rng (f | F1()) c= F2() and
A14: for x being set st x in (F1() \/ {F3()}) \ F1() holds
f . x = H1(x) ;
 f . F3() = F4() by A4, A14;
hence  x in  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & f . F3() = F4() )  }  by A11, A12, A13; ::_thesis: verum
end;
A15:  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & f . F3() = F4() )  }  c=  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
f . x = H1(x) ) )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & f . F3() = F4() )  }  or x in  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
f . x = H1(x) ) )  }  )
assume  x in  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & f . F3() = F4() )  }  ; ::_thesis: x in  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
f . x = H1(x) ) )  }
then consider f being Function of (F1() \/ {F3()}),(F2() \/ {F4()}) such that
A16: x = f and
A17: P1[f,F1() \/ {F3()},F2() \/ {F4()}] and
A18: rng (f | F1()) c= F2() and
A19: f . F3() = F4() ;
 for x being set st x in (F1() \/ {F3()}) \ F1() holds
H1(x) = f . x by A4, A19;
hence  x in  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
f . x = H1(x) ) )  }  by A16, A17, A18; ::_thesis: verum
end;
A20: ( F1() c= F1() \/ {F3()} & F2() c= F2() \/ {F4()} ) by XBOOLE_1:7;
 card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & ( for x being set st x in (F1() \/ {F3()}) \ F1() holds
f . x = H1(x) ) )  }  from STIRL2_1:sch_3(A8, A20, A1, A7);
hence  card  {  f where f is Function of F1(),F2() : P1[f,F1(),F2()]  }  = card  {  f where f is Function of (F1() \/ {F3()}),(F2() \/ {F4()}) : ( P1[f,F1() \/ {F3()},F2() \/ {F4()}] & rng (f | F1()) c= F2() & f . F3() = F4() )  }  by A10, A15, XBOOLE_0:def_10; ::_thesis: verum
end;

theorem Th34: :: STIRL2_1:34
for n, k being Nat
for f being Function of (n + 1),(k + 1) st f is onto & f is "increasing & f " {(f . n)} = {n} holds
f . n = k
proof
let n, k be Nat; ::_thesis: for f being Function of (n + 1),(k + 1) st f is onto & f is "increasing & f " {(f . n)} = {n} holds
f . n = k
let f be Function of (n + 1),(k + 1); ::_thesis: ( f is onto & f is "increasing & f " {(f . n)} = {n} implies f . n = k )
assume that
A1: ( f is onto & f is "increasing ) and
A2: f " {(f . n)} = {n} ; ::_thesis: f . n = k
assume A3: f . n <> k ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( f . n > k or f . n < k ) by A3, XXREAL_0:1;
supposeA4: f . n > k ; ::_thesis: contradiction
 f . n < k + 1 by NAT_1:44;
hence  contradiction by A4, NAT_1:13; ::_thesis: verum
end;
supposeA5: f . n < k ; ::_thesis: contradiction
A6: min* (f " {k}) <= (n + 1) - 1 by Th16;
A7: rng f = k + 1 by A1, FUNCT_2:def_3;
 k < k + 1 by NAT_1:13;
then  k in rng f by A7, NAT_1:44;
then  ( min* (f " {(f . n)}) < min* (f " {k}) & k in NAT & n in NAT ) by A1, A5, A7, Def1, ORDINAL1:def_12;
hence  contradiction by A2, A6, Th5; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;

theorem Th35: :: STIRL2_1:35
for n, k being Nat
for f being Function of (n + 1),k st k <> 0 & f " {(f . n)} <> {n} holds
ex m being Nat st
( m in f " {(f . n)} & m <> n )
proof
let n, k be Nat; ::_thesis: for f being Function of (n + 1),k st k <> 0 & f " {(f . n)} <> {n} holds
ex m being Nat st
( m in f " {(f . n)} & m <> n )
let f be Function of (n + 1),k; ::_thesis: ( k <> 0 & f " {(f . n)} <> {n} implies ex m being Nat st
( m in f " {(f . n)} & m <> n ) )
assume that
A1: k <> 0 and
A2: f " {(f . n)} <> {n} ; ::_thesis: ex m being Nat st
( m in f " {(f . n)} & m <> n )
A3: n < n + 1 by NAT_1:13;
A4: f . n in {(f . n)} by TARSKI:def_1;
 dom f = n + 1 by A1, FUNCT_2:def_1;
then  n in dom f by A3, NAT_1:44;
then  n in f " {(f . n)} by A4, FUNCT_1:def_7;
then  ex m being set st
( m in f " {(f . n)} & m <> n ) by A2, ZFMISC_1:35;
hence  ex m being Nat st
( m in f " {(f . n)} & m <> n ) ; ::_thesis: verum
end;

theorem Th36: :: STIRL2_1:36
for n, k, m, l being Nat
for f being Function of n,k
for g being Function of (n + m),(k + l) st g is "increasing & f = g | n holds
for i, j being Nat st i in rng f & j in rng f & i < j holds
min* (f " {i}) < min* (f " {j})
proof
let n, k, m, l be Nat; ::_thesis: for f being Function of n,k
for g being Function of (n + m),(k + l) st g is "increasing & f = g | n holds
for i, j being Nat st i in rng f & j in rng f & i < j holds
min* (f " {i}) < min* (f " {j})
let f be Function of n,k; ::_thesis: for g being Function of (n + m),(k + l) st g is "increasing & f = g | n holds
for i, j being Nat st i in rng f & j in rng f & i < j holds
min* (f " {i}) < min* (f " {j})
let g be Function of (n + m),(k + l); ::_thesis: ( g is "increasing & f = g | n implies for i, j being Nat st i in rng f & j in rng f & i < j holds
min* (f " {i}) < min* (f " {j}) )
assume that
A1: g is "increasing and
A2: f = g | n ; ::_thesis: for i, j being Nat st i in rng f & j in rng f & i < j holds
min* (f " {i}) < min* (f " {j})
let i, j be Nat; ::_thesis: ( i in rng f & j in rng f & i < j implies min* (f " {i}) < min* (f " {j}) )
assume that
A3: i in rng f and
A4: j in rng f and
A5: i < j ; ::_thesis: min* (f " {i}) < min* (f " {j})
A6: for k1 being Element of NAT st k1 in rng f holds
( k1 in rng g & min* (f " {k1}) = min* (g " {k1}) )
proof
A7: n is Subset of NAT by Th8;
let k1 be Element of NAT ; ::_thesis: ( k1 in rng f implies ( k1 in rng g & min* (f " {k1}) = min* (g " {k1}) ) )
assume A8: k1 in rng f ; ::_thesis: ( k1 in rng g & min* (f " {k1}) = min* (g " {k1}) )
consider x being set  such that
A9: x in dom f and
A10: f . x = k1 by A8, FUNCT_1:def_3;
A11: dom f = n by A8, FUNCT_2:def_1;
 x in n by A9;
then reconsider x9 = x as Element of NAT by A7;
A12: x9 < n by A9, NAT_1:44;
A13: f . x9 = g . x9 by A2, A9, FUNCT_1:47;
A14: dom g = n + m by A8, FUNCT_2:def_1;
 n <= n + m by NAT_1:11;
then A15: n c= n + m by NAT_1:39;
A16: now__::_thesis:_for_n1_being_Nat_st_n1_in_f_"_{k1}_holds_
min*_(g_"_{k1})_<=_n1
let n1 be Nat; ::_thesis: ( n1 in f " {k1} implies min* (g " {k1}) <= n1 )
assume A17: n1 in f " {k1} ; ::_thesis: min* (g " {k1}) <= n1
A18: n1 in n by A11, A17, FUNCT_1:def_7;
 f . n1 in {k1} by A17, FUNCT_1:def_7;
then  g . n1 in {k1} by A2, A11, A18, FUNCT_1:47;
then  n1 in g " {k1} by A14, A15, A18, FUNCT_1:def_7;
hence  min* (g " {k1}) <= n1 by NAT_1:def_1; ::_thesis: verum
end;
 k1 in {k1} by TARSKI:def_1;
then A19: x9 in g " {k1} by A9, A10, A11, A14, A15, A13, FUNCT_1:def_7;
then  min* (g " {k1}) <= x9 by NAT_1:def_1;
then  min* (g " {k1}) < n by A12, XXREAL_0:2;
then A20: min* (g " {k1}) in dom f by A11, NAT_1:44;
 min* (g " {k1}) in g " {k1} by A19, NAT_1:def_1;
then  g . (min* (g " {k1})) in {k1} by FUNCT_1:def_7;
then  f . (min* (g " {k1})) in {k1} by A2, A20, FUNCT_1:47;
then  min* (g " {k1}) in f " {k1} by A20, FUNCT_1:def_7;
hence  ( k1 in rng g & min* (f " {k1}) = min* (g " {k1}) ) by A9, A10, A11, A14, A15, A13, A16, FUNCT_1:def_3, NAT_1:def_1; ::_thesis: verum
end;
A21: ( i in NAT & j in NAT ) by ORDINAL1:def_12;
then A22: j in rng g by A4, A6;
A23: min* (f " {j}) = min* (g " {j}) by A4, A6, A21;
A24: min* (f " {i}) = min* (g " {i}) by A3, A6, A21;
 i in rng g by A3, A6, A21;
hence  min* (f " {i}) < min* (f " {j}) by A1, A5, A22, A24, A23, Def1; ::_thesis: verum
end;

theorem Th37: :: STIRL2_1:37
for n, k being Nat
for f being Function of (n + 1),(k + 1) st f is onto & f is "increasing & f " {(f . n)} = {n} holds
( rng (f | n) c= k & ( for g being Function of n,k st g = f | n holds
( g is onto & g is "increasing ) ) )
proof
let n, k be Nat; ::_thesis: for f being Function of (n + 1),(k + 1) st f is onto & f is "increasing & f " {(f . n)} = {n} holds
( rng (f | n) c= k & ( for g being Function of n,k st g = f | n holds
( g is onto & g is "increasing ) ) )
let f be Function of (n + 1),(k + 1); ::_thesis: ( f is onto & f is "increasing & f " {(f . n)} = {n} implies ( rng (f | n) c= k & ( for g being Function of n,k st g = f | n holds
( g is onto & g is "increasing ) ) ) )
assume that
A1: ( f is onto & f is "increasing ) and
A2: f " {(f . n)} = {n} ; ::_thesis: ( rng (f | n) c= k & ( for g being Function of n,k st g = f | n holds
( g is onto & g is "increasing ) ) )
now__::_thesis:_(_rng_(f_|_n)_c=_k_&_(_for_g_being_Function_of_n,k_st_g_=_f_|_n_holds_
(_g_is_onto_&_g_is_"increasing_)_)_)
percases ( n = 0 or n > 0 ) ;
supposeA3: n = 0 ; ::_thesis: ( rng (f | n) c= k & ( for g being Function of n,k st g = f | n holds
( g is onto & g is "increasing ) ) )
then  0 + 1 >= k + 1 by A1, Th17;
then  k = 0 by XREAL_1:6;
hence  ( rng (f | n) c= k & ( for g being Function of n,k st g = f | n holds
( g is onto & g is "increasing ) ) ) by A3, Th15; ::_thesis: verum
end;
supposeA4: n > 0 ; ::_thesis: ( rng (f | n) c= k & ( for g being Function of n,k st g = f | n holds
( g is onto & g is "increasing ) ) )
thus A5: rng (f | n) c= k ::_thesis: for g being Function of n,k st g = f | n holds
( g is onto & g is "increasing )
proof
let fi be set ; :: according to TARSKI:def_3 ::_thesis: ( not fi in rng (f | n) or fi in k )
assume A6: fi in rng (f | n) ; ::_thesis: fi in k
 rng (f | n) c= rng f by RELAT_1:70;
then  fi in rng f by A6;
then A7: fi in k + 1 ;
 k + 1 is Subset of NAT by Th8;
then reconsider fi = fi as Element of NAT by A7;
consider i being set  such that
A8: i in dom (f | n) and
A9: (f | n) . i = fi by A6, FUNCT_1:def_3;
 i in (dom f) /\ n by A8, RELAT_1:61;
then A10: i in n by XBOOLE_0:def_4;
 n is Subset of NAT by Th8;
then reconsider i = i as Element of NAT by A10;
A11: f . i < k
proof
 f . i < k + 1 by NAT_1:44;
then A12: f . i <= k by NAT_1:13;
assume  f . i >= k ; ::_thesis: contradiction
then A13: f . i = k by A12, XXREAL_0:1;
A14: f . i in {(f . i)} by TARSKI:def_1;
A15: f . n = k by A1, A2, Th34;
A16: i in (dom f) /\ n by A8, RELAT_1:61;
then  i in dom f by XBOOLE_0:def_4;
then  i in f " {(f . n)} by A13, A14, A15, FUNCT_1:def_7;
then  ( i >= min* (f " {(f . n)}) & i in NAT & n in NAT ) by NAT_1:def_1, ORDINAL1:def_12;
then A17: i >= n by A2, Th5;
 i in n by A16, XBOOLE_0:def_4;
hence  contradiction by A17, NAT_1:44; ::_thesis: verum
end;
 f . i = (f | n) . i by A8, FUNCT_1:47;
hence  fi in k by A9, A11, NAT_1:44; ::_thesis: verum
end;
thus  for g being Function of n,k st g = f | n holds
( g is onto & g is "increasing ) ::_thesis: verum
proof
let g be Function of n,k; ::_thesis: ( g = f | n implies ( g is onto & g is "increasing ) )
assume A18: g = f | n ; ::_thesis: ( g is onto & g is "increasing )
 k c= rng g
proof
 n < n + 1 by NAT_1:13;
then A19: n c= n + 1 by NAT_1:39;
 dom f = n + 1 by FUNCT_2:def_1;
then A20: n = (dom f) /\ n by A19, XBOOLE_1:28;
let k1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not k1 in k or k1 in rng g )
assume A21: k1 in k ; ::_thesis: k1 in rng g
 k is Subset of NAT by Th8;
then reconsider k9 = k1 as Element of NAT by A21;
 k9 < k by A21, NAT_1:44;
then  k9 < k + 1 by NAT_1:13;
then A22: k9 in k + 1 by NAT_1:44;
A23: dom f = n + 1 by FUNCT_2:def_1;
 rng f = k + 1 by A1, FUNCT_2:def_3;
then consider n1 being set  such that
A24: n1 in dom f and
A25: f . n1 = k9 by A22, FUNCT_1:def_3;
 n + 1 is Subset of NAT by Th8;
then reconsider n1 = n1 as Element of NAT by A24, A23;
 n1 < n + 1 by A24, NAT_1:44;
then A26: n1 <= n by NAT_1:13;
 f . n = k by A1, A2, Th34;
then  n1 <> n by A21, A25;
then A27: n1 < n by A26, XXREAL_0:1;
 (dom f) /\ n = dom (f | n) by RELAT_1:61;
then A28: n1 in dom g by A18, A27, A20, NAT_1:44;
then  g . n1 in rng g by FUNCT_1:def_3;
hence  k1 in rng g by A18, A25, A28, FUNCT_1:47; ::_thesis: verum
end;
then  k = rng g by XBOOLE_0:def_10;
hence  g is onto by FUNCT_2:def_3; ::_thesis: g is "increasing
A29: (dom f) /\ n = dom (f | n) by RELAT_1:61;
 n < n + 1 by NAT_1:13;
then A30: n c= n + 1 by NAT_1:39;
 dom f = n + 1 by FUNCT_2:def_1;
then A31: n = (dom f) /\ n by A30, XBOOLE_1:28;
 0 in n by A4, NAT_1:44;
then  (f | n) . 0 in rng (f | n) by A31, A29, FUNCT_1:def_3;
then A32: ( n = 0 iff k = 0 ) by A5;
 for i, j being Nat st i in rng g & j in rng g & i < j holds
min* (g " {i}) < min* (g " {j}) by A1, A18, Th36;
hence  g is "increasing by A32, Def1; ::_thesis: verum
end;
end;
end;
end;
hence  ( rng (f | n) c= k & ( for g being Function of n,k st g = f | n holds
( g is onto & g is "increasing ) ) ) ; ::_thesis: verum
end;

theorem Th38: :: STIRL2_1:38
for n, k being Nat
for f being Function of (n + 1),k
for g being Function of n,k st f is onto & f is "increasing & f " {(f . n)} <> {n} & f | n = g holds
( g is onto & g is "increasing )
proof
let n, k be Nat; ::_thesis: for f being Function of (n + 1),k
for g being Function of n,k st f is onto & f is "increasing & f " {(f . n)} <> {n} & f | n = g holds
( g is onto & g is "increasing )
let f be Function of (n + 1),k; ::_thesis: for g being Function of n,k st f is onto & f is "increasing & f " {(f . n)} <> {n} & f | n = g holds
( g is onto & g is "increasing )
let g be Function of n,k; ::_thesis: ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f | n = g implies ( g is onto & g is "increasing ) )
assume that
A1: ( f is onto & f is "increasing ) and
A2: f " {(f . n)} <> {n} and
A3: f | n = g ; ::_thesis: ( g is onto & g is "increasing )
now__::_thesis:_(_g_is_onto_&_g_is_"increasing_)
percases ( k = 0 or k > 0 ) ;
suppose k = 0 ; ::_thesis: ( g is onto & g is "increasing )
hence  ( g is onto & g is "increasing ) by A1, Def1; ::_thesis: verum
end;
supposeA4: k > 0 ; ::_thesis: ( g is onto & g is "increasing )
A5: rng f = k by A1, FUNCT_2:def_3;
now__::_thesis:_(_g_is_onto_&_g_is_"increasing_)
 k = k + 0 ;
then A6: for i, j being Nat st i in rng g & j in rng g & i < j holds
min* (g " {i}) < min* (g " {j}) by A1, A3, Th36;
A7: k c= rng g
proof
A8: n + 1 is Subset of NAT by Th8;
let k1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not k1 in k or k1 in rng g )
assume A9: k1 in k ; ::_thesis: k1 in rng g
consider x being set  such that
A10: x in dom f and
A11: f . x = k1 by A5, A9, FUNCT_1:def_3;
 dom f = n + 1 by A9, FUNCT_2:def_1;
then reconsider x = x as Element of NAT by A10, A8;
 x < n + 1 by A10, NAT_1:44;
then A12: x <= n by NAT_1:13;
now__::_thesis:_k1_in_rng_g
percases ( x < n or x = n ) by A12, XXREAL_0:1;
supposeA13: x < n ; ::_thesis: k1 in rng g
A14: dom g = n by A4, FUNCT_2:def_1;
A15: x in n by A13, NAT_1:44;
then  g . x = f . x by A3, A14, FUNCT_1:47;
hence  k1 in rng g by A11, A15, A14, FUNCT_1:def_3; ::_thesis: verum
end;
suppose x = n ; ::_thesis: k1 in rng g
then consider m being Nat such that
A16: m in f " {k1} and
A17: m <> n by A2, A4, A11, Th35;
 f . m in {k1} by A16, FUNCT_1:def_7;
then A18: f . m = k1 by TARSKI:def_1;
 m in dom f by A16, FUNCT_1:def_7;
then  m < n + 1 by NAT_1:44;
then  m <= n by NAT_1:13;
then  m < n by A17, XXREAL_0:1;
then A19: m in n by NAT_1:44;
A20: n = dom g by A4, FUNCT_2:def_1;
then  g . m = f . m by A3, A19, FUNCT_1:47;
hence  k1 in rng g by A19, A20, A18, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
hence  k1 in rng g ; ::_thesis: verum
end;
then A21: rng g = k by XBOOLE_0:def_10;
 ( n = 0 iff k = 0 ) by A4, A7;
hence  ( g is onto & g is "increasing ) by A21, A6, Def1, FUNCT_2:def_3; ::_thesis: verum
end;
hence  ( g is onto & g is "increasing ) ; ::_thesis: verum
end;
end;
end;
hence  ( g is onto & g is "increasing ) ; ::_thesis: verum
end;

theorem Th39: :: STIRL2_1:39
for n, k, m being Nat
for f being Function of n,k
for g being Function of (n + 1),(k + m) st f is onto & f is "increasing & f = g | n holds
for i, j being Nat st i in rng g & j in rng g & i < j holds
min* (g " {i}) < min* (g " {j})
proof
let n, k, m be Nat; ::_thesis: for f being Function of n,k
for g being Function of (n + 1),(k + m) st f is onto & f is "increasing & f = g | n holds
for i, j being Nat st i in rng g & j in rng g & i < j holds
min* (g " {i}) < min* (g " {j})
let f be Function of n,k; ::_thesis: for g being Function of (n + 1),(k + m) st f is onto & f is "increasing & f = g | n holds
for i, j being Nat st i in rng g & j in rng g & i < j holds
min* (g " {i}) < min* (g " {j})
let g be Function of (n + 1),(k + m); ::_thesis: ( f is onto & f is "increasing & f = g | n implies for i, j being Nat st i in rng g & j in rng g & i < j holds
min* (g " {i}) < min* (g " {j}) )
assume that
A1: ( f is onto & f is "increasing ) and
A2: f = g | n ; ::_thesis: for i, j being Nat st i in rng g & j in rng g & i < j holds
min* (g " {i}) < min* (g " {j})
A3: for i being Nat st i < n holds
f . i = g . i
proof
 ( k = 0 iff n = 0 ) by A1, Def1;
then A4: dom f = n by FUNCT_2:def_1;
let i be Nat; ::_thesis: ( i < n implies f . i = g . i )
assume  i < n ; ::_thesis: f . i = g . i
then  i in n by NAT_1:44;
hence  f . i = g . i by A2, A4, FUNCT_1:47; ::_thesis: verum
end;
A5: for l being Nat st l in rng g & not l in rng f holds
l = g . n
proof
A6: n + 1 is Subset of NAT by Th8;
let l be Nat; ::_thesis: ( l in rng g & not l in rng f implies l = g . n )
assume that
A7: l in rng g and
A8: not l in rng f ; ::_thesis: l = g . n
consider x being set  such that
A9: x in dom g and
A10: g . x = l by A7, FUNCT_1:def_3;
assume A11: l <> g . n ; ::_thesis: contradiction
 dom g = n + 1 by A7, FUNCT_2:def_1;
then reconsider x = x as Element of NAT by A9, A6;
 x < n + 1 by A9, NAT_1:44;
then  x <= n by NAT_1:13;
then A12: x < n by A11, A10, XXREAL_0:1;
then A13: x in n by NAT_1:44;
 k <> 0 by A1, A12, Def1;
then A14: dom f = n by FUNCT_2:def_1;
 f . x = g . x by A3, A12;
hence  contradiction by A8, A10, A14, A13, FUNCT_1:def_3; ::_thesis: verum
end;
A15: for l being Nat st l in rng g & not l in rng f holds
min* (g " {l}) = n
proof
A16: n < n + 1 by NAT_1:13;
let l be Nat; ::_thesis: ( l in rng g & not l in rng f implies min* (g " {l}) = n )
assume that
A17: l in rng g and
A18: not l in rng f ; ::_thesis: min* (g " {l}) = n
A19: l in {l} by TARSKI:def_1;
 dom g = n + 1 by A17, FUNCT_2:def_1;
then A20: n in dom g by A16, NAT_1:44;
 g . n = l by A5, A17, A18;
then  n in g " {l} by A20, A19, FUNCT_1:def_7;
then  min* (g " {l}) in g " {l} by NAT_1:def_1;
then A21: g . (min* (g " {l})) in {l} by FUNCT_1:def_7;
assume A22: min* (g " {l}) <> n ; ::_thesis: contradiction
 min* (g " {l}) <= (n + 1) - 1 by Th16;
then A23: min* (g " {l}) < n by A22, XXREAL_0:1;
then  k <> 0 by A1, Def1;
then A24: dom f = n by FUNCT_2:def_1;
 min* (g " {l}) in n by A23, NAT_1:44;
then A25: f . (min* (g " {l})) in rng f by A24, FUNCT_1:def_3;
 f . (min* (g " {l})) = g . (min* (g " {l})) by A3, A23;
hence  contradiction by A18, A21, A25, TARSKI:def_1; ::_thesis: verum
end;
A26: for k1 being Element of NAT st k1 in rng f holds
min* (g " {k1}) = min* (f " {k1})
proof
 n <= n + 1 by NAT_1:11;
then A27: n c= n + 1 by NAT_1:39;
let k1 be Element of NAT ; ::_thesis: ( k1 in rng f implies min* (g " {k1}) = min* (f " {k1}) )
assume A28: k1 in rng f ; ::_thesis: min* (g " {k1}) = min* (f " {k1})
consider x being set  such that
A29: x in dom f and
A30: f . x = k1 by A28, FUNCT_1:def_3;
A31: x in n by A29;
A32: dom g = n + 1 by A28, FUNCT_2:def_1;
 n is Subset of NAT by Th8;
then reconsider x = x as Element of NAT by A31;
 k1 in {k1} by TARSKI:def_1;
then A33: x in f " {k1} by A29, A30, FUNCT_1:def_7;
then A34: min* (f " {k1}) <= x by NAT_1:def_1;
A35: x < n by A29, NAT_1:44;
then A36: min* (f " {k1}) < n by A34, XXREAL_0:2;
A37: dom f = n by A28, FUNCT_2:def_1;
A38: now__::_thesis:_for_n1_being_Nat_st_n1_in_g_"_{k1}_holds_
min*_(f_"_{k1})_<=_n1
let n1 be Nat; ::_thesis: ( n1 in g " {k1} implies min* (f " {k1}) <= n1 )
assume A39: n1 in g " {k1} ; ::_thesis: min* (f " {k1}) <= n1
 n1 in n + 1 by A32, A39, FUNCT_1:def_7;
then  n1 < n + 1 by NAT_1:44;
then A40: n1 <= n by NAT_1:13;
now__::_thesis:_min*_(f_"_{k1})_<=_n1
percases ( n1 < n or n1 = n ) by A40, XXREAL_0:1;
supposeA41: n1 < n ; ::_thesis: min* (f " {k1}) <= n1
 g . n1 in {k1} by A39, FUNCT_1:def_7;
then A42: f . n1 in {k1} by A3, A41;
 n1 in dom f by A37, A41, NAT_1:44;
then  n1 in f " {k1} by A42, FUNCT_1:def_7;
hence  min* (f " {k1}) <= n1 by NAT_1:def_1; ::_thesis: verum
end;
suppose n1 = n ; ::_thesis: min* (f " {k1}) <= n1
hence  min* (f " {k1}) <= n1 by A34, A35, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  min* (f " {k1}) <= n1 ; ::_thesis: verum
end;
 min* (f " {k1}) in f " {k1} by A33, NAT_1:def_1;
then  f . (min* (f " {k1})) in {k1} by FUNCT_1:def_7;
then A43: g . (min* (f " {k1})) in {k1} by A3, A36;
 min* (f " {k1}) in n by A36, NAT_1:44;
then  min* (f " {k1}) in g " {k1} by A32, A27, A43, FUNCT_1:def_7;
hence  min* (g " {k1}) = min* (f " {k1}) by A38, NAT_1:def_1; ::_thesis: verum
end;
let i, j be Nat; ::_thesis: ( i in rng g & j in rng g & i < j implies min* (g " {i}) < min* (g " {j}) )
assume that
A44: i in rng g and
A45: j in rng g and
A46: i < j ; ::_thesis: min* (g " {i}) < min* (g " {j})
A47: for l being Nat st l in rng g & not l in rng f holds
l >= k
proof
let l be Nat; ::_thesis: ( l in rng g & not l in rng f implies l >= k )
assume that
 l in rng g and
A48: not l in rng f ; ::_thesis: l >= k
assume  l < k ; ::_thesis: contradiction
then  l in k by NAT_1:44;
hence  contradiction by A1, A48, FUNCT_2:def_3; ::_thesis: verum
end;
A49: ( i in NAT & j in NAT ) by ORDINAL1:def_12;
now__::_thesis:_min*_(g_"_{i})_<_min*_(g_"_{j})
percases ( ( i in rng f & j in rng f ) or ( i in rng f & not j in rng f ) or ( not i in rng f & j in rng f ) or ( not i in rng f & not j in rng f ) ) ;
supposeA50: ( i in rng f & j in rng f ) ; ::_thesis: min* (g " {i}) < min* (g " {j})
then A51: min* (g " {j}) = min* (f " {j}) by A26, A49;
 min* (g " {i}) = min* (f " {i}) by A26, A50, A49;
hence  min* (g " {i}) < min* (g " {j}) by A1, A46, A50, A51, Def1; ::_thesis: verum
end;
supposeA52: ( i in rng f & not j in rng f ) ; ::_thesis: min* (g " {i}) < min* (g " {j})
then A53: n <> 0 ;
then  min* (f " {i}) <= n - 1 by Th16;
then A54: min* (g " {i}) <= n - 1 by A26, A52, A49;
 n - 1 is Element of NAT by A53, NAT_1:20;
then A55: n - 1 < (n - 1) + 1 by NAT_1:13;
 min* (g " {j}) = n by A45, A15, A52;
hence  min* (g " {i}) < min* (g " {j}) by A54, A55, XXREAL_0:2; ::_thesis: verum
end;
supposeA56: ( not i in rng f & j in rng f ) ; ::_thesis: min* (g " {i}) < min* (g " {j})
then A57: j < k by NAT_1:44;
 i >= k by A44, A47, A56;
hence  min* (g " {i}) < min* (g " {j}) by A46, A57, XXREAL_0:2; ::_thesis: verum
end;
supposeA58: ( not i in rng f & not j in rng f ) ; ::_thesis: min* (g " {i}) < min* (g " {j})
then  i = g . n by A44, A5;
hence  min* (g " {i}) < min* (g " {j}) by A45, A46, A5, A58; ::_thesis: verum
end;
end;
end;
hence  min* (g " {i}) < min* (g " {j}) ; ::_thesis: verum
end;

theorem Th40: :: STIRL2_1:40
for n, k being Nat
for f being Function of n,k
for g being Function of (n + 1),(k + 1) st f is onto & f is "increasing & f = g | n & g . n = k holds
( g is onto & g is "increasing & g " {(g . n)} = {n} )
proof
let n, k be Nat; ::_thesis: for f being Function of n,k
for g being Function of (n + 1),(k + 1) st f is onto & f is "increasing & f = g | n & g . n = k holds
( g is onto & g is "increasing & g " {(g . n)} = {n} )
let f be Function of n,k; ::_thesis: for g being Function of (n + 1),(k + 1) st f is onto & f is "increasing & f = g | n & g . n = k holds
( g is onto & g is "increasing & g " {(g . n)} = {n} )
let g be Function of (n + 1),(k + 1); ::_thesis: ( f is onto & f is "increasing & f = g | n & g . n = k implies ( g is onto & g is "increasing & g " {(g . n)} = {n} ) )
assume that
A1: ( f is onto & f is "increasing ) and
A2: f = g | n and
A3: g . n = k ; ::_thesis: ( g is onto & g is "increasing & g " {(g . n)} = {n} )
 k + 1 c= rng g
proof
let x9 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x9 in k + 1 or x9 in rng g )
assume A4: x9 in k + 1 ; ::_thesis: x9 in rng g
 k + 1 is Subset of NAT by Th8;
then reconsider x = x9 as Element of NAT by A4;
 x < k + 1 by A4, NAT_1:44;
then A5: x <= k by NAT_1:13;
now__::_thesis:_x9_in_rng_g
percases ( x < k or x = k ) by A5, XXREAL_0:1;
supposeA6: x < k ; ::_thesis: x9 in rng g
A7: rng f = k by A1, FUNCT_2:def_3;
 x in k by A6, NAT_1:44;
then consider y being set  such that
A8: y in dom f and
A9: f . y = x by A7, FUNCT_1:def_3;
A10: dom g = n + 1 by FUNCT_2:def_1;
 ( n = 0 iff k = 0 ) by A1, Def1;
then A11: dom f = n by FUNCT_2:def_1;
 n <= n + 1 by NAT_1:11;
then A12: n c= n + 1 by NAT_1:39;
 f . y = g . y by A2, A8, FUNCT_1:47;
hence  x9 in rng g by A8, A9, A11, A12, A10, FUNCT_1:def_3; ::_thesis: verum
end;
supposeA13: x = k ; ::_thesis: x9 in rng g
 n < n + 1 by NAT_1:13;
then A14: n in n + 1 by NAT_1:44;
 dom g = n + 1 by FUNCT_2:def_1;
hence  x9 in rng g by A3, A13, A14, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
hence  x9 in rng g ; ::_thesis: verum
end;
then  k + 1 = rng g by XBOOLE_0:def_10;
hence  g is onto by FUNCT_2:def_3; ::_thesis: ( g is "increasing & g " {(g . n)} = {n} )
 for i, j being Nat st i in rng g & j in rng g & i < j holds
min* (g " {i}) < min* (g " {j}) by A1, A2, Th39;
hence  g is "increasing by Def1; ::_thesis: g " {(g . n)} = {n}
thus  g " {(g . n)} = {n} ::_thesis: verum
proof
assume  g " {(g . n)} <> {n} ; ::_thesis: contradiction
then consider m being Nat such that
A15: m in g " {(g . n)} and
A16: m <> n by Th35;
 g . m in {(g . n)} by A15, FUNCT_1:def_7;
then A17: g . m = k by A3, TARSKI:def_1;
 m in dom g by A15, FUNCT_1:def_7;
then  m < n + 1 by NAT_1:44;
then  m <= n by NAT_1:13;
then A18: m < n by A16, XXREAL_0:1;
 ( n = 0 iff k = 0 ) by A1, Def1;
then  dom f = n by FUNCT_2:def_1;
then A19: m in dom f by A18, NAT_1:44;
then A20: f . m in rng f by FUNCT_1:def_3;
 f . m = g . m by A2, A19, FUNCT_1:47;
hence  contradiction by A20, A17, NAT_1:44; ::_thesis: verum
end;
end;

theorem Th41: :: STIRL2_1:41
for n, k being Nat
for f being Function of n,k
for g being Function of (n + 1),k st f is onto & f is "increasing & f = g | n & g . n < k holds
( g is onto & g is "increasing & g " {(g . n)} <> {n} )
proof
let n, k be Nat; ::_thesis: for f being Function of n,k
for g being Function of (n + 1),k st f is onto & f is "increasing & f = g | n & g . n < k holds
( g is onto & g is "increasing & g " {(g . n)} <> {n} )
let f be Function of n,k; ::_thesis: for g being Function of (n + 1),k st f is onto & f is "increasing & f = g | n & g . n < k holds
( g is onto & g is "increasing & g " {(g . n)} <> {n} )
let g be Function of (n + 1),k; ::_thesis: ( f is onto & f is "increasing & f = g | n & g . n < k implies ( g is onto & g is "increasing & g " {(g . n)} <> {n} ) )
assume that
A1: ( f is onto & f is "increasing ) and
A2: f = g | n and
A3: g . n < k ; ::_thesis: ( g is onto & g is "increasing & g " {(g . n)} <> {n} )
 k = rng f by A1, FUNCT_2:def_3;
then consider x being set  such that
A4: x in dom f and
A5: f . x = g . n by A3, FUNCT_1:def_3;
 g . n = g . x by A2, A4, A5, FUNCT_1:47;
then A6: g . x in {(g . n)} by TARSKI:def_1;
 k c= rng g
proof
 n <= n + 1 by NAT_1:13;
then A7: n c= n + 1 by NAT_1:39;
 ( n = 0 iff k = 0 ) by A1, Def1;
then A8: dom f = n by FUNCT_2:def_1;
let x9 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x9 in k or x9 in rng g )
assume A9: x9 in k ; ::_thesis: x9 in rng g
 k is Subset of NAT by Th8;
then reconsider x = x9 as Element of NAT by A9;
 rng f = k by A1, FUNCT_2:def_3;
then consider y being set  such that
A10: y in dom f and
A11: f . y = x by A9, FUNCT_1:def_3;
A12: dom g = n + 1 by A3, FUNCT_2:def_1;
 f . y = g . y by A2, A10, FUNCT_1:47;
hence  x9 in rng g by A10, A11, A8, A7, A12, FUNCT_1:def_3; ::_thesis: verum
end;
then  k = rng g by XBOOLE_0:def_10;
hence  g is onto by FUNCT_2:def_3; ::_thesis: ( g is "increasing & g " {(g . n)} <> {n} )
 k = k + 0 ;
then  for i, j being Nat st i in rng g & j in rng g & i < j holds
min* (g " {i}) < min* (g " {j}) by A1, A2, Th39;
hence  g is "increasing by A3, Def1; ::_thesis: g " {(g . n)} <> {n}
 n <= n + 1 by NAT_1:11;
then A13: n c= n + 1 by NAT_1:39;
A14: x in n by A4;
then A15: x <> n ;
 dom g = n + 1 by A3, FUNCT_2:def_1;
then  x in g " {(g . n)} by A14, A13, A6, FUNCT_1:def_7;
hence  g " {(g . n)} <> {n} by A15, TARSKI:def_1; ::_thesis: verum
end;

Lm1:  for k, n being Nat st k < n holds
n \/ {k} = n
proof
let k, n be Nat; ::_thesis: ( k < n implies n \/ {k} = n )
assume  k < n ; ::_thesis: n \/ {k} = n
then  k in n by NAT_1:44;
then  {k} c= n by ZFMISC_1:31;
hence  n \/ {k} = n by XBOOLE_1:12; ::_thesis: verum
end;

theorem Th42: :: STIRL2_1:42
for n, k being Nat holds card  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
proof
let n, k be Nat; ::_thesis: card  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
set F1 =  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  ;
set F2 =  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  ;
now__::_thesis:_card__{__f_where_f_is_Function_of_(n_+_1),(k_+_1)_:_(_f_is_onto_&_f_is_"increasing_&_f_"_{(f_._n)}_=_{n}_)__}__=_card__{__f_where_f_is_Function_of_n,k_:_(_f_is_onto_&_f_is_"increasing_)__}_
percases not ( not ( k = 0 & n <> 0 ) & k = 0 & not n = 0 ) ;
supposeA1: ( k = 0 & n <> 0 ) ; ::_thesis: card  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
A2:  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  is empty
proof
assume  not  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  is empty ; ::_thesis: contradiction
then consider x being set  such that
A3: x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  by XBOOLE_0:def_1;
consider f being Function of (n + 1),(k + 1) such that
 x = f and
 ( f is onto & f is "increasing ) and
A4: f " {(f . n)} = {n} by A3;
 0 in n + 1 by NAT_1:44;
then A5: 0 in dom f by FUNCT_2:def_1;
A6: 0 in {0} by TARSKI:def_1;
A7: f . 0 = 0 by A1, CARD_1:49, TARSKI:def_1;
 f . n = 0 by A1, CARD_1:49, TARSKI:def_1;
then  0 in {n} by A4, A7, A5, A6, FUNCT_1:def_7;
hence  contradiction by A1, TARSKI:def_1; ::_thesis: verum
end;
 n block k = 0 by A1, Th31;
hence  card  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  by A2; ::_thesis: verum
end;
supposeA8: ( k = 0 implies n = 0 ) ; ::_thesis: card  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
defpred S1[ set , set , set ] means  for i, j being Nat st i = $2 & j = $3 holds
ex f being Function of i,j st
( f = $1 & f is onto & f is "increasing & ( n < i implies f " {(f . n)} = {n} ) );
A9: not n in n ;
set FF2 =  {  f where f is Function of n,k : S1[f,n,k]  }  ;
set FF1 =  {  f where f is Function of (n \/ {n}),(k \/ {k}) : ( S1[f,n \/ {n},k \/ {k}] & rng (f | n) c= k & f . n = k )  }  ;
A10: for f being Function of (n \/ {n}),(k \/ {k}) st f . n = k holds
( S1[f,n \/ {n},k \/ {k}] iff S1[f | n,n,k] )
proof
let f9 be Function of (n \/ {n}),(k \/ {k}); ::_thesis: ( f9 . n = k implies ( S1[f9,n \/ {n},k \/ {k}] iff S1[f9 | n,n,k] ) )
assume A11: f9 . n = k ; ::_thesis: ( S1[f9,n \/ {n},k \/ {k}] iff S1[f9 | n,n,k] )
thus  ( S1[f9,n \/ {n},k \/ {k}] implies S1[f9 | n,n,k] ) ::_thesis: ( S1[f9 | n,n,k] implies S1[f9,n \/ {n},k \/ {k}] )
proof
 n <= n + 1 by NAT_1:11;
then A12: n c= n + 1 by NAT_1:39;
A13: n + 1 = n \/ {n} by AFINSQ_1:2;
A14: k + 1 = k \/ {k} by AFINSQ_1:2;
assume  S1[f9,n \/ {n},k \/ {k}] ; ::_thesis: S1[f9 | n,n,k]
then consider f being Function of (n + 1),(k + 1) such that
A15: f = f9 and
A16: ( f is onto & f is "increasing ) and
A17: ( n < n + 1 implies f " {(f . n)} = {n} ) by A13, A14;
A18: rng (f | n) c= k by A16, A17, Th37, NAT_1:13;
A19: dom (f | n) = (dom f) /\ n by RELAT_1:61;
 dom f = n + 1 by FUNCT_2:def_1;
then  dom (f | n) = n by A12, A19, XBOOLE_1:28;
then reconsider fn = f | n as Function of n,k by A18, FUNCT_2:2;
let i, j be Nat; ::_thesis: ( i = n & j = k implies ex f being Function of i,j st
( f = f9 | n & f is onto & f is "increasing & ( n < i implies f " {(f . n)} = {n} ) ) )
assume that
A20: i = n and
A21: j = k ; ::_thesis: ex f being Function of i,j st
( f = f9 | n & f is onto & f is "increasing & ( n < i implies f " {(f . n)} = {n} ) )
reconsider fi = fn as Function of i,j by A20, A21;
 ( fi is onto & fi is "increasing ) by A16, A17, A20, A21, Th37, NAT_1:13;
hence  ex f being Function of i,j st
( f = f9 | n & f is onto & f is "increasing & ( n < i implies f " {(f . n)} = {n} ) ) by A15, A20; ::_thesis: verum
end;
thus  ( S1[f9 | n,n,k] implies S1[f9,n \/ {n},k \/ {k}] ) ::_thesis: verum
proof
 n \/ {n} = n + 1 by AFINSQ_1:2;
then reconsider f = f9 as Function of (n + 1),(k + 1) by AFINSQ_1:2;
assume  S1[f9 | n,n,k] ; ::_thesis: S1[f9,n \/ {n},k \/ {k}]
then A22: ex fn being Function of n,k st
( fn = f9 | n & fn is onto & fn is "increasing & ( n < n implies fn " {(fn . n)} = {n} ) ) ;
let i, j be Nat; ::_thesis: ( i = n \/ {n} & j = k \/ {k} implies ex f being Function of i,j st
( f = f9 & f is onto & f is "increasing & ( n < i implies f " {(f . n)} = {n} ) ) )
assume that
A23: i = n \/ {n} and
A24: j = k \/ {k} ; ::_thesis: ex f being Function of i,j st
( f = f9 & f is onto & f is "increasing & ( n < i implies f " {(f . n)} = {n} ) )
reconsider f1 = f as Function of i,j by A23, A24;
A25: ( n < i implies f1 " {(f1 . n)} = {n} ) by A11, A22, Th40;
A26: k + 1 = j by A24, AFINSQ_1:2;
 n + 1 = i by A23, AFINSQ_1:2;
then  ( f1 is onto & f1 is "increasing ) by A11, A22, A26, Th40;
hence  ex f being Function of i,j st
( f = f9 & f is onto & f is "increasing & ( n < i implies f " {(f . n)} = {n} ) ) by A25; ::_thesis: verum
end;
end;
A27: ( k is empty implies n is empty ) by A8;
A28: card  {  f where f is Function of n,k : S1[f,n,k]  }  = card  {  f where f is Function of (n \/ {n}),(k \/ {k}) : ( S1[f,n \/ {n},k \/ {k}] & rng (f | n) c= k & f . n = k )  }  from STIRL2_1:sch_4(A27, A9, A10);
A29:  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  c=  {  f where f is Function of n,k : S1[f,n,k]  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  or x in  {  f where f is Function of n,k : S1[f,n,k]  }  )
assume  x in  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  ; ::_thesis: x in  {  f where f is Function of n,k : S1[f,n,k]  }
then A30: ex f being Function of n,k st
( x = f & f is onto & f is "increasing ) ;
then  S1[x,n,k] ;
hence  x in  {  f where f is Function of n,k : S1[f,n,k]  }  by A30; ::_thesis: verum
end;
A31:  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  c=  {  f where f is Function of (n \/ {n}),(k \/ {k}) : ( S1[f,n \/ {n},k \/ {k}] & rng (f | n) c= k & f . n = k )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  or x in  {  f where f is Function of (n \/ {n}),(k \/ {k}) : ( S1[f,n \/ {n},k \/ {k}] & rng (f | n) c= k & f . n = k )  }  )
assume  x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  ; ::_thesis: x in  {  f where f is Function of (n \/ {n}),(k \/ {k}) : ( S1[f,n \/ {n},k \/ {k}] & rng (f | n) c= k & f . n = k )  }
then consider f being Function of (n + 1),(k + 1) such that
A32: f = x and
A33: ( f is onto & f is "increasing ) and
A34: f " {(f . n)} = {n} ;
A35: rng (f | n) c= k by A33, A34, Th37;
A36: S1[f,n \/ {n},k \/ {k}]
proof
let i, j be Nat; ::_thesis: ( i = n \/ {n} & j = k \/ {k} implies ex f being Function of i,j st
( f = f & f is onto & f is "increasing & ( n < i implies f " {(f . n)} = {n} ) ) )
assume that
A37: i = n \/ {n} and
A38: j = k \/ {k} ; ::_thesis: ex f being Function of i,j st
( f = f & f is onto & f is "increasing & ( n < i implies f " {(f . n)} = {n} ) )
A39: j = k + 1 by A38, AFINSQ_1:2;
 i = n + 1 by A37, AFINSQ_1:2;
hence  ex f being Function of i,j st
( f = f & f is onto & f is "increasing & ( n < i implies f " {(f . n)} = {n} ) ) by A33, A34, A39; ::_thesis: verum
end;
A40: k + 1 = k \/ {k} by AFINSQ_1:2;
A41: n + 1 = n \/ {n} by AFINSQ_1:2;
 f . n = k by A33, A34, Th34;
hence  x in  {  f where f is Function of (n \/ {n}),(k \/ {k}) : ( S1[f,n \/ {n},k \/ {k}] & rng (f | n) c= k & f . n = k )  }  by A32, A36, A35, A41, A40; ::_thesis: verum
end;
A42:  {  f where f is Function of n,k : S1[f,n,k]  }  c=  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of n,k : S1[f,n,k]  }  or x in  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  )
assume  x in  {  f where f is Function of n,k : S1[f,n,k]  }  ; ::_thesis: x in  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
then consider f being Function of n,k such that
A43: x = f and
A44: S1[f,n,k] ;
 ex g being Function of n,k st
( g = f & g is onto & g is "increasing & ( n < n implies g " {(g . n)} = {n} ) ) by A44;
hence  x in  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  by A43; ::_thesis: verum
end;
  {  f where f is Function of (n \/ {n}),(k \/ {k}) : ( S1[f,n \/ {n},k \/ {k}] & rng (f | n) c= k & f . n = k )  }  c=  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of (n \/ {n}),(k \/ {k}) : ( S1[f,n \/ {n},k \/ {k}] & rng (f | n) c= k & f . n = k )  }  or x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  )
assume  x in  {  f where f is Function of (n \/ {n}),(k \/ {k}) : ( S1[f,n \/ {n},k \/ {k}] & rng (f | n) c= k & f . n = k )  }  ; ::_thesis: x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }
then consider f being Function of (n \/ {n}),(k \/ {k}) such that
A45: x = f and
A46: S1[f,n \/ {n},k \/ {k}] and
 rng (f | n) c= k and
 f . n = k ;
A47: k + 1 = k \/ {k} by AFINSQ_1:2;
 n + 1 = n \/ {n} by AFINSQ_1:2;
then  ex f9 being Function of (n + 1),(k + 1) st
( f = f9 & f9 is onto & f9 is "increasing & ( n < n + 1 implies f9 " {(f9 . n)} = {n} ) ) by A46, A47;
hence  x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  by A45, NAT_1:13; ::_thesis: verum
end;
then   {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  =  {  f where f is Function of (n \/ {n}),(k \/ {k}) : ( S1[f,n \/ {n},k \/ {k}] & rng (f | n) c= k & f . n = k )  }  by A31, XBOOLE_0:def_10;
hence  card  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  by A28, A29, A42, XBOOLE_0:def_10; ::_thesis: verum
end;
end;
end;
hence  card  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  ; ::_thesis: verum
end;

theorem Th43: :: STIRL2_1:43
for k, n, l being Nat st l < k holds
card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
proof
let k, n, l be Nat; ::_thesis: ( l < k implies card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  )
assume A1: l < k ; ::_thesis: card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
set F2 =  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  ;
set F1 =  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  ;
now__::_thesis:_card__{__f_where_f_is_Function_of_(n_+_1),k_:_(_f_is_onto_&_f_is_"increasing_&_f_"_{(f_._n)}_<>_{n}_&_f_._n_=_l_)__}__=_card__{__f_where_f_is_Function_of_n,k_:_(_f_is_onto_&_f_is_"increasing_)__}_
percases not ( not ( k = 0 & n <> 0 ) & k = 0 & not n = 0 ) ;
suppose ( k = 0 & n <> 0 ) ; ::_thesis: card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
hence  card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  by A1; ::_thesis: verum
end;
supposeA2: ( k = 0 implies n = 0 ) ; ::_thesis: card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
defpred S1[ set , set , set ] means  for i, j being Nat st i = $2 & j = $3 holds
ex f being Function of i,j st
( f = $1 & f is onto & f is "increasing & ( n < i implies f " {(f . n)} <> {n} ) );
A3: not n in n ;
set FF2 =  {  f where f is Function of n,k : S1[f,n,k]  }  ;
set FF1 =  {  f where f is Function of (n \/ {n}),(k \/ {l}) : ( S1[f,n \/ {n},k \/ {l}] & rng (f | n) c= k & f . n = l )  }  ;
A4: for f being Function of (n \/ {n}),(k \/ {l}) st f . n = l holds
( S1[f,n \/ {n},k \/ {l}] iff S1[f | n,n,k] )
proof
let f9 be Function of (n \/ {n}),(k \/ {l}); ::_thesis: ( f9 . n = l implies ( S1[f9,n \/ {n},k \/ {l}] iff S1[f9 | n,n,k] ) )
assume A5: f9 . n = l ; ::_thesis: ( S1[f9,n \/ {n},k \/ {l}] iff S1[f9 | n,n,k] )
thus  ( S1[f9,n \/ {n},k \/ {l}] implies S1[f9 | n,n,k] ) ::_thesis: ( S1[f9 | n,n,k] implies S1[f9,n \/ {n},k \/ {l}] )
proof
 n <= n + 1 by NAT_1:13;
then A6: n c= n + 1 by NAT_1:39;
assume A7: S1[f9,n \/ {n},k \/ {l}] ; ::_thesis: S1[f9 | n,n,k]
A8: n + 1 = n \/ {n} by AFINSQ_1:2;
 k = k \/ {l} by A1, Lm1;
then consider f being Function of (n + 1),k such that
A9: f = f9 and
A10: ( f is onto & f is "increasing ) and
A11: ( n < n + 1 implies f " {(f . n)} <> {n} ) by A7, A8;
A12: dom (f | n) = (dom f) /\ n by RELAT_1:61;
A13: rng (f | n) c= k ;
 dom f = n + 1 by A1, FUNCT_2:def_1;
then  dom (f | n) = n by A6, A12, XBOOLE_1:28;
then reconsider fn = f | n as Function of n,k by A13, FUNCT_2:2;
let i, j be Nat; ::_thesis: ( i = n & j = k implies ex f being Function of i,j st
( f = f9 | n & f is onto & f is "increasing & ( n < i implies f " {(f . n)} <> {n} ) ) )
assume that
A14: i = n and
A15: j = k ; ::_thesis: ex f being Function of i,j st
( f = f9 | n & f is onto & f is "increasing & ( n < i implies f " {(f . n)} <> {n} ) )
reconsider fi = fn as Function of i,j by A14, A15;
 ( fi is onto & fi is "increasing ) by A10, A11, A14, A15, Th38, NAT_1:13;
hence  ex f being Function of i,j st
( f = f9 | n & f is onto & f is "increasing & ( n < i implies f " {(f . n)} <> {n} ) ) by A9, A14; ::_thesis: verum
end;
thus  ( S1[f9 | n,n,k] implies S1[f9,n \/ {n},k \/ {l}] ) ::_thesis: verum
proof
 n \/ {n} = n + 1 by AFINSQ_1:2;
then reconsider f = f9 as Function of (n + 1),k by A1, Lm1;
assume  S1[f9 | n,n,k] ; ::_thesis: S1[f9,n \/ {n},k \/ {l}]
then A16: ex fn being Function of n,k st
( fn = f9 | n & fn is onto & fn is "increasing & ( n < n implies fn " {(fn . n)} <> {n} ) ) ;
let i, j be Nat; ::_thesis: ( i = n \/ {n} & j = k \/ {l} implies ex f being Function of i,j st
( f = f9 & f is onto & f is "increasing & ( n < i implies f " {(f . n)} <> {n} ) ) )
assume that
A17: i = n \/ {n} and
A18: j = k \/ {l} ; ::_thesis: ex f being Function of i,j st
( f = f9 & f is onto & f is "increasing & ( n < i implies f " {(f . n)} <> {n} ) )
reconsider f1 = f as Function of i,j by A17, A18;
A19: ( n < i implies f1 " {(f1 . n)} <> {n} ) by A1, A5, A16, Th41;
A20: n + 1 = i by A17, AFINSQ_1:2;
 k = j by A1, A18, Lm1;
then  ( f1 is onto & f1 is "increasing ) by A1, A5, A16, A20, Th41;
hence  ex f being Function of i,j st
( f = f9 & f is onto & f is "increasing & ( n < i implies f " {(f . n)} <> {n} ) ) by A19; ::_thesis: verum
end;
end;
A21: ( k is empty implies n is empty ) by A2;
A22: card  {  f where f is Function of n,k : S1[f,n,k]  }  = card  {  f where f is Function of (n \/ {n}),(k \/ {l}) : ( S1[f,n \/ {n},k \/ {l}] & rng (f | n) c= k & f . n = l )  }  from STIRL2_1:sch_4(A21, A3, A4);
A23:  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  c=  {  f where f is Function of n,k : S1[f,n,k]  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  or x in  {  f where f is Function of n,k : S1[f,n,k]  }  )
assume  x in  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  ; ::_thesis: x in  {  f where f is Function of n,k : S1[f,n,k]  }
then A24: ex f being Function of n,k st
( x = f & f is onto & f is "increasing ) ;
then  S1[x,n,k] ;
hence  x in  {  f where f is Function of n,k : S1[f,n,k]  }  by A24; ::_thesis: verum
end;
A25:  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  c=  {  f where f is Function of (n \/ {n}),(k \/ {l}) : ( S1[f,n \/ {n},k \/ {l}] & rng (f | n) c= k & f . n = l )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  or x in  {  f where f is Function of (n \/ {n}),(k \/ {l}) : ( S1[f,n \/ {n},k \/ {l}] & rng (f | n) c= k & f . n = l )  }  )
assume  x in  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  ; ::_thesis: x in  {  f where f is Function of (n \/ {n}),(k \/ {l}) : ( S1[f,n \/ {n},k \/ {l}] & rng (f | n) c= k & f . n = l )  }
then consider f being Function of (n + 1),k such that
A26: f = x and
A27: ( f is onto & f is "increasing ) and
A28: f " {(f . n)} <> {n} and
A29: f . n = l ;
A30: S1[f,n \/ {n},k \/ {l}]
proof
let i, j be Nat; ::_thesis: ( i = n \/ {n} & j = k \/ {l} implies ex f being Function of i,j st
( f = f & f is onto & f is "increasing & ( n < i implies f " {(f . n)} <> {n} ) ) )
assume that
A31: i = n \/ {n} and
A32: j = k \/ {l} ; ::_thesis: ex f being Function of i,j st
( f = f & f is onto & f is "increasing & ( n < i implies f " {(f . n)} <> {n} ) )
A33: i = n + 1 by A31, AFINSQ_1:2;
 j = k by A1, A32, Lm1;
hence  ex f being Function of i,j st
( f = f & f is onto & f is "increasing & ( n < i implies f " {(f . n)} <> {n} ) ) by A27, A28, A33; ::_thesis: verum
end;
A34: k = k \/ {l} by A1, Lm1;
A35: n + 1 = n \/ {n} by AFINSQ_1:2;
 rng (f | n) c= k ;
hence  x in  {  f where f is Function of (n \/ {n}),(k \/ {l}) : ( S1[f,n \/ {n},k \/ {l}] & rng (f | n) c= k & f . n = l )  }  by A26, A29, A30, A35, A34; ::_thesis: verum
end;
A36:  {  f where f is Function of n,k : S1[f,n,k]  }  c=  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of n,k : S1[f,n,k]  }  or x in  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  )
assume  x in  {  f where f is Function of n,k : S1[f,n,k]  }  ; ::_thesis: x in  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }
then consider f being Function of n,k such that
A37: x = f and
A38: S1[f,n,k] ;
 ex g being Function of n,k st
( g = f & g is onto & g is "increasing & ( n < n implies g " {(g . n)} <> {n} ) ) by A38;
hence  x in  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  by A37; ::_thesis: verum
end;
  {  f where f is Function of (n \/ {n}),(k \/ {l}) : ( S1[f,n \/ {n},k \/ {l}] & rng (f | n) c= k & f . n = l )  }  c=  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }
proof
A39: n + 1 = n \/ {n} by AFINSQ_1:2;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of (n \/ {n}),(k \/ {l}) : ( S1[f,n \/ {n},k \/ {l}] & rng (f | n) c= k & f . n = l )  }  or x in  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  )
assume  x in  {  f where f is Function of (n \/ {n}),(k \/ {l}) : ( S1[f,n \/ {n},k \/ {l}] & rng (f | n) c= k & f . n = l )  }  ; ::_thesis: x in  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }
then consider f being Function of (n \/ {n}),(k \/ {l}) such that
A40: x = f and
A41: S1[f,n \/ {n},k \/ {l}] and
 rng (f | n) c= k and
A42: f . n = l ;
 k = k \/ {l} by A1, Lm1;
then  ex f9 being Function of (n + 1),k st
( f = f9 & f9 is onto & f9 is "increasing & ( n < n + 1 implies f9 " {(f9 . n)} <> {n} ) ) by A41, A39;
hence  x in  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  by A40, A42, NAT_1:13; ::_thesis: verum
end;
then   {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  =  {  f where f is Function of (n \/ {n}),(k \/ {l}) : ( S1[f,n \/ {n},k \/ {l}] & rng (f | n) c= k & f . n = l )  }  by A25, XBOOLE_0:def_10;
hence  card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  by A22, A23, A36, XBOOLE_0:def_10; ::_thesis: verum
end;
end;
end;
hence  card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} & f . n = l )  }  = card  {  f where f is Function of n,k : ( f is onto & f is "increasing )  }  ; ::_thesis: verum
end;

theorem Th44: :: STIRL2_1:44
for f being Function
for n being Nat holds (union (rng (f | n))) \/ (f . n) = union (rng (f | (n + 1)))
proof
let f be Function; ::_thesis: for n being Nat holds (union (rng (f | n))) \/ (f . n) = union (rng (f | (n + 1)))
let n be Nat; ::_thesis: (union (rng (f | n))) \/ (f . n) = union (rng (f | (n + 1)))
thus  (union (rng (f | n))) \/ (f . n) c= union (rng (f | (n + 1))) :: according to XBOOLE_0:def_10 ::_thesis: union (rng (f | (n + 1))) c= (union (rng (f | n))) \/ (f . n)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (union (rng (f | n))) \/ (f . n) or x in union (rng (f | (n + 1))) )
assume A1: x in (union (rng (f | n))) \/ (f . n) ; ::_thesis: x in union (rng (f | (n + 1)))
now__::_thesis:_x_in_union_(rng_(f_|_(n_+_1)))
percases ( x in union (rng (f | n)) or x in f . n ) by A1, XBOOLE_0:def_3;
suppose x in union (rng (f | n)) ; ::_thesis: x in union (rng (f | (n + 1)))
then consider Y being set  such that
A2: x in Y and
A3: Y in rng (f | n) by TARSKI:def_4;
consider X being set  such that
A4: X in dom (f | n) and
A5: (f | n) . X = Y by A3, FUNCT_1:def_3;
A6: (f | n) . X = f . X by A4, FUNCT_1:47;
 n <= n + 1 by NAT_1:11;
then  n c= n + 1 by NAT_1:39;
then A7: (dom f) /\ n c= (dom f) /\ (n + 1) by XBOOLE_1:26;
 X in (dom f) /\ n by A4, RELAT_1:61;
then  X in (dom f) /\ (n + 1) by A7;
then A8: X in dom (f | (n + 1)) by RELAT_1:61;
then A9: (f | (n + 1)) . X = f . X by FUNCT_1:47;
 (f | (n + 1)) . X in rng (f | (n + 1)) by A8, FUNCT_1:def_3;
hence  x in union (rng (f | (n + 1))) by A2, A5, A9, A6, TARSKI:def_4; ::_thesis: verum
end;
supposeA10: x in f . n ; ::_thesis: x in union (rng (f | (n + 1)))
 n < n + 1 by NAT_1:13;
then A11: n in n + 1 by NAT_1:44;
 n in dom f by A10, FUNCT_1:def_2;
then  n in (dom f) /\ (n + 1) by A11, XBOOLE_0:def_4;
then A12: n in dom (f | (n + 1)) by RELAT_1:61;
then A13: (f | (n + 1)) . n = f . n by FUNCT_1:47;
 (f | (n + 1)) . n in rng (f | (n + 1)) by A12, FUNCT_1:def_3;
hence  x in union (rng (f | (n + 1))) by A10, A13, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
hence  x in union (rng (f | (n + 1))) ; ::_thesis: verum
end;
thus  union (rng (f | (n + 1))) c= (union (rng (f | n))) \/ (f . n) ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (rng (f | (n + 1))) or x in (union (rng (f | n))) \/ (f . n) )
assume  x in union (rng (f | (n + 1))) ; ::_thesis: x in (union (rng (f | n))) \/ (f . n)
then consider Y being set  such that
A14: x in Y and
A15: Y in rng (f | (n + 1)) by TARSKI:def_4;
consider X being set  such that
A16: X in dom (f | (n + 1)) and
A17: (f | (n + 1)) . X = Y by A15, FUNCT_1:def_3;
A18: X in (dom f) /\ (n + 1) by A16, RELAT_1:61;
then A19: X in n + 1 by XBOOLE_0:def_4;
A20: X in dom f by A18, XBOOLE_0:def_4;
 n + 1 is Subset of NAT by Th8;
then reconsider X = X as Element of NAT by A19;
 X < n + 1 by A19, NAT_1:44;
then A21: X <= n by NAT_1:13;
now__::_thesis:_x_in_(union_(rng_(f_|_n)))_\/_(f_._n)
percases ( X < n or X = n ) by A21, XXREAL_0:1;
suppose X < n ; ::_thesis: x in (union (rng (f | n))) \/ (f . n)
then  X in n by NAT_1:44;
then  X in (dom f) /\ n by A20, XBOOLE_0:def_4;
then A22: X in dom (f | n) by RELAT_1:61;
then A23: (f | n) . X in rng (f | n) by FUNCT_1:def_3;
A24: f . X = (f | (n + 1)) . X by A16, FUNCT_1:47;
 (f | n) . X = f . X by A22, FUNCT_1:47;
then  x in union (rng (f | n)) by A14, A17, A24, A23, TARSKI:def_4;
hence  x in (union (rng (f | n))) \/ (f . n) by XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA25: X = n ; ::_thesis: x in (union (rng (f | n))) \/ (f . n)
 f . X = (f | (n + 1)) . X by A16, FUNCT_1:47;
hence  x in (union (rng (f | n))) \/ (f . n) by A14, A17, A25, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
hence  x in (union (rng (f | n))) \/ (f . n) ; ::_thesis: verum
end;
end;

scheme :: STIRL2_1:sch 5
Sch6{ F1() ->  non empty set , F2() ->  Nat, P1[ set , set ] } :
 ex p being XFinSequence of F1() st
( dom p = F2() & ( for k being Nat st k in F2() holds
P1[k,p . k] ) )
provided
A1: for k being Nat st k in F2() holds
ex x being Element of F1() st P1[k,x]
proof
A2: for k being set st k in F2() holds
ex x being set st
( x in F1() & P1[k,x] )
proof
let k be set ; ::_thesis: ( k in F2() implies ex x being set st
( x in F1() & P1[k,x] ) )
assume A3: k in F2() ; ::_thesis: ex x being set st
( x in F1() & P1[k,x] )
 F2() is Subset of NAT by Th8;
then reconsider k9 = k as Element of NAT by A3;
 ex x being Element of F1() st P1[k9,x] by A1, A3;
hence  ex x being set st
( x in F1() & P1[k,x] ) ; ::_thesis: verum
end;
consider f being Function of F2(),F1() such that
A4: for x being set st x in F2() holds
P1[x,f . x] from FUNCT_2:sch_1(A2);
 dom f = F2() by FUNCT_2:def_1;
then reconsider p = f as XFinSequence of F1() by AFINSQ_1:5;
take  p ; ::_thesis: ( dom p = F2() & ( for k being Nat st k in F2() holds
P1[k,p . k] ) )
thus  ( dom p = F2() & ( for k being Nat st k in F2() holds
P1[k,p . k] ) ) by A4, FUNCT_2:def_1; ::_thesis: verum
end;

Lm2: now__::_thesis:_for_D_being_non_empty_set_
for_F_being_XFinSequence_of_D_st_(_for_i_being_Nat_st_i_in_dom_F_holds_
F_._i_is_finite_)_&_(_for_i,_j_being_Nat_st_i_in_dom_F_&_j_in_dom_F_&_i_<>_j_holds_
F_._i_misses_F_._j_)_holds_
ex_CardF_being_XFinSequence_of_NAT_st_
(_dom_CardF_=_dom_F_&_(_for_i_being_Nat_st_i_in_dom_CardF_holds_
CardF_._i_=_card_(F_._i)_)_&_card_(union_(rng_F))_=_Sum_CardF_)
let D be non empty set ; ::_thesis: for F being XFinSequence of D st ( for i being Nat st i in dom F holds
F . i is finite ) & ( for i, j being Nat st i in dom F & j in dom F & i <> j holds
F . i misses F . j ) holds
ex CardF being XFinSequence of NAT st
( dom CardF = dom F & ( for i being Nat st i in dom CardF holds
CardF . i = card (F . i) ) & card (union (rng F)) = Sum CardF )
let F be XFinSequence of D; ::_thesis: ( ( for i being Nat st i in dom F holds
F . i is finite ) & ( for i, j being Nat st i in dom F & j in dom F & i <> j holds
F . i misses F . j ) implies ex CardF being XFinSequence of NAT st
( dom CardF = dom F & ( for i being Nat st i in dom CardF holds
CardF . i = card (F . i) ) & card (union (rng F)) = Sum CardF ) )
assume that
A1: for i being Nat st i in dom F holds
F . i is finite and
A2: for i, j being Nat st i in dom F & j in dom F & i <> j holds
F . i misses F . j ; ::_thesis: ex CardF being XFinSequence of NAT st
( dom CardF = dom F & ( for i being Nat st i in dom CardF holds
CardF . i = card (F . i) ) & card (union (rng F)) = Sum CardF )
thus  ex CardF being XFinSequence of NAT st
( dom CardF = dom F & ( for i being Nat st i in dom CardF holds
CardF . i = card (F . i) ) & card (union (rng F)) = Sum CardF ) ::_thesis: verum
proof
defpred S1[ Nat, set ] means $2 = card (F . $1);
A3: for k being Nat st k in len F holds
ex x being Element of NAT st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in len F implies ex x being Element of NAT st S1[k,x] )
assume  k in len F ; ::_thesis: ex x being Element of NAT st S1[k,x]
then consider m being Nat such that
A4: F . k,m are_equipotent by A1, CARD_1:43;
 card (F . k) = card m by A4, CARD_1:5;
hence  ex x being Element of NAT st S1[k,x] ; ::_thesis: verum
end;
consider CardF being XFinSequence of NAT  such that
A5: ( dom CardF = len F & ( for k being Nat st k in len F holds
S1[k,CardF . k] ) ) from STIRL2_1:sch_5(A3);
take  CardF ; ::_thesis: ( dom CardF = dom F & ( for i being Nat st i in dom CardF holds
CardF . i = card (F . i) ) & card (union (rng F)) = Sum CardF )
thus  dom CardF = dom F by A5; ::_thesis: ( ( for i being Nat st i in dom CardF holds
CardF . i = card (F . i) ) & card (union (rng F)) = Sum CardF )
thus  for i being Nat st i in dom CardF holds
CardF . i = card (F . i) by A5; ::_thesis: card (union (rng F)) = Sum CardF
A6: addnat "**" CardF = Sum CardF by AFINSQ_2:51;
percases ( len CardF = 0 or len CardF <> 0 ) ;
supposeA7: len CardF = 0 ; ::_thesis: card (union (rng F)) = Sum CardF
then  union (rng F) is empty by A5, RELAT_1:42, ZFMISC_1:2;
hence  card (union (rng F)) = Sum CardF by A7, A6, AFINSQ_2:def_8, BINOP_2:5; ::_thesis: verum
end;
supposeA8: len CardF <> 0 ; ::_thesis: card (union (rng F)) = Sum CardF
then consider f being Function of NAT,NAT such that
A9: f . 0 = CardF . 0 and
A10: for n being Nat st n + 1 < len CardF holds
f . (n + 1) = addnat . ((f . n),(CardF . (n + 1))) and
A11: addnat "**" CardF = f . ((len CardF) - 1) by AFINSQ_2:def_8;
defpred S2[ Nat] means ( $1 < len CardF implies ( card (union (rng (F | ($1 + 1)))) = f . $1 & union (rng (F | ($1 + 1))) is finite ) );
A12: for k being Nat st S2[k] holds
S2[k + 1]
proof
let k be Nat; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A13: S2[k] ; ::_thesis: S2[k + 1]
set k1 = k + 1;
set Fk1 = F | (k + 1);
set rFk1 = rng (F | (k + 1));
assume A14: k + 1 < len CardF ; ::_thesis: ( card (union (rng (F | ((k + 1) + 1)))) = f . (k + 1) & union (rng (F | ((k + 1) + 1))) is finite )
reconsider urFk1 = union (rng (F | (k + 1))) as finite set by A13, A14, NAT_1:13;
A15: f . (k + 1) = addnat . ((f . k),(CardF . (k + 1))) by A10, A14;
A16: union (rng (F | (k + 1))) misses F . (k + 1)
proof
assume  union (rng (F | (k + 1))) meets F . (k + 1) ; ::_thesis: contradiction
then consider x being set  such that
A17: x in (union (rng (F | (k + 1)))) /\ (F . (k + 1)) by XBOOLE_0:4;
A18: x in F . (k + 1) by A17, XBOOLE_0:def_4;
A19: k + 1 in dom F by A5, A14, NAT_1:44;
 x in union (rng (F | (k + 1))) by A17, XBOOLE_0:def_4;
then consider Y being set  such that
A20: x in Y and
A21: Y in rng (F | (k + 1)) by TARSKI:def_4;
consider X being set  such that
A22: X in dom (F | (k + 1)) and
A23: (F | (k + 1)) . X = Y by A21, FUNCT_1:def_3;
reconsider X = X as Element of NAT by A22;
A24: (F | (k + 1)) . X = F . X by A22, FUNCT_1:47;
A25: X in (dom F) /\ (k + 1) by A22, RELAT_1:61;
then  X in k + 1 by XBOOLE_0:def_4;
then A26: X <> k + 1 ;
 X in dom F by A25, XBOOLE_0:def_4;
then  Y misses F . (k + 1) by A2, A23, A19, A26, A24;
hence  contradiction by A20, A18, XBOOLE_0:3; ::_thesis: verum
end;
 k + 1 in dom F by A5, A14, NAT_1:44;
then reconsider Fk1 = F . (k + 1) as finite set by A1;
 k + 1 in len F by A5, A14, NAT_1:44;
then  card Fk1 = CardF . (k + 1) by A5;
then A27: f . (k + 1) = (f . k) + (card Fk1) by A15, BINOP_2:def_23;
 card (urFk1 \/ Fk1) = (f . k) + (card Fk1) by A13, A14, A16, CARD_2:40, NAT_1:13;
hence  ( card (union (rng (F | ((k + 1) + 1)))) = f . (k + 1) & union (rng (F | ((k + 1) + 1))) is finite ) by A27, Th44; ::_thesis: verum
end;
reconsider C1 = (len CardF) - 1 as Element of NAT by A8, NAT_1:20;
A28: C1 < C1 + 1 by NAT_1:13;
A29: F | (len CardF) = F by A5, RELAT_1:68;
A30: S2[ 0 ]
proof
assume  0 < len CardF ; ::_thesis: ( card (union (rng (F | (0 + 1)))) = f . 0 & union (rng (F | (0 + 1))) is finite )
A31: union (rng (F | (0 + 1))) = (union (rng (F | 0))) \/ (F . 0) by Th44;
 0 in len CardF by A8, NAT_1:44;
hence  ( card (union (rng (F | (0 + 1)))) = f . 0 & union (rng (F | (0 + 1))) is finite ) by A1, A5, A9, A31, ZFMISC_1:2; ::_thesis: verum
end;
 for k being Nat holds S2[k] from NAT_1:sch_2(A30, A12);
hence  card (union (rng F)) = Sum CardF by A11, A28, A29, A6; ::_thesis: verum
end;
end;
end;
end;

scheme :: STIRL2_1:sch 6
Sch8{ F1() ->  finite set , F2() ->  finite set , F3() ->  set , P1[ set ], F4() ->  Function of (card F2()),F2() } :
 ex F being XFinSequence of NAT st
( dom F = card F2() & card  {  g where g is Function of F1(),F2() : P1[g]  }  = Sum F & ( for i being Nat st i in dom F holds
F . i = card  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . i )  }  ) )
provided
A1: ( F4() is onto & F4() is one-to-one ) and
A2: not F2() is empty and
A3: F3() in F1()
proof
defpred S1[ Nat, set ] means $2 =  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . $1 )  }  ;
consider n being Nat such that
A4: F2(),n are_equipotent by CARD_1:43;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A5: for k being Nat st k in n holds
ex x being Subset of (Funcs (F1(),F2())) st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in n implies ex x being Subset of (Funcs (F1(),F2())) st S1[k,x] )
assume  k in n ; ::_thesis: ex x being Subset of (Funcs (F1(),F2())) st S1[k,x]
set F0 =  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . k )  }  ;
  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . k )  }  c= Funcs (F1(),F2())
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . k )  }  or x in Funcs (F1(),F2()) )
assume  x in  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . k )  }  ; ::_thesis: x in Funcs (F1(),F2())
then  ex g being Function of F1(),F2() st
( x = g & P1[g] & g . F3() = F4() . k ) ;
hence  x in Funcs (F1(),F2()) by A2, FUNCT_2:8; ::_thesis: verum
end;
hence  ex x being Subset of (Funcs (F1(),F2())) st S1[k,x] ; ::_thesis: verum
end;
consider F being XFinSequence of bool (Funcs (F1(),F2())) such that
A6: ( dom F = n & ( for k being Nat st k in n holds
S1[k,F . k] ) ) from STIRL2_1:sch_5(A5);
A7: for i, j being Nat st i in dom F & j in dom F & i <> j holds
F . i misses F . j
proof
let i, j be Nat; ::_thesis: ( i in dom F & j in dom F & i <> j implies F . i misses F . j )
assume that
A8: i in dom F and
A9: j in dom F and
A10: i <> j ; ::_thesis: F . i misses F . j
assume  F . i meets F . j ; ::_thesis: contradiction
then consider x being set  such that
A11: x in (F . i) /\ (F . j) by XBOOLE_0:4;
 x in F . i by A11, XBOOLE_0:def_4;
then  x in  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . i )  }  by A6, A8;
then A12: ex gi being Function of F1(),F2() st
( x = gi & P1[gi] & gi . F3() = F4() . i ) ;
A13: card F2() = card n by A4, CARD_1:5;
A14: card n = n by CARD_1:def_2;
 x in F . j by A11, XBOOLE_0:def_4;
then  x in  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . j )  }  by A6, A9;
then A15: ex gj being Function of F1(),F2() st
( x = gj & P1[gj] & gj . F3() = F4() . j ) ;
 dom F4() = card F2() by A2, FUNCT_2:def_1;
hence  contradiction by A1, A6, A8, A9, A10, A12, A15, A13, A14, FUNCT_1:def_4; ::_thesis: verum
end;
A16: for i being Nat st i in dom F holds
F . i is finite
proof
let i be Nat; ::_thesis: ( i in dom F implies F . i is finite )
assume  i in dom F ; ::_thesis: F . i is finite
then A17: F . i in rng F by FUNCT_1:def_3;
A18: Funcs (F1(),F2()) is finite by FRAENKEL:6;
thus  F . i is finite by A17, A18; ::_thesis: verum
end;
consider CardF being XFinSequence of NAT  such that
A19: dom CardF = dom F and
A20: for i being Nat st i in dom CardF holds
CardF . i = card (F . i) and
A21: card (union (rng F)) = Sum CardF by Lm2, A16, A7;
take  CardF ; ::_thesis: ( dom CardF = card F2() & card  {  g where g is Function of F1(),F2() : P1[g]  }  = Sum CardF & ( for i being Nat st i in dom CardF holds
CardF . i = card  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . i )  }  ) )
thus  dom CardF = card F2() by A4, A6, A19, CARD_1:def_2; ::_thesis: ( card  {  g where g is Function of F1(),F2() : P1[g]  }  = Sum CardF & ( for i being Nat st i in dom CardF holds
CardF . i = card  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . i )  }  ) )
thus  card  {  g where g is Function of F1(),F2() : P1[g]  }  = Sum CardF ::_thesis: for i being Nat st i in dom CardF holds
CardF . i = card  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . i )  }
proof
set G =  {  g where g is Function of F1(),F2() : P1[g]  }  ;
A22:  {  g where g is Function of F1(),F2() : P1[g]  }  c= union (rng F)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  g where g is Function of F1(),F2() : P1[g]  }  or x in union (rng F) )
assume  x in  {  g where g is Function of F1(),F2() : P1[g]  }  ; ::_thesis: x in union (rng F)
then consider g being Function of F1(),F2() such that
A23: x = g and
A24: P1[g] ;
A25: card n = n by CARD_1:def_2;
A26: rng F4() = F2() by A1, FUNCT_2:def_3;
A27: card F2() = card n by A4, CARD_1:5;
 dom g = F1() by A2, FUNCT_2:def_1;
then  g . F3() in rng g by A3, FUNCT_1:def_3;
then consider y being set  such that
A28: y in dom F4() and
A29: F4() . y = g . F3() by A26, FUNCT_1:def_3;
 F . y =  {  g1 where g1 is Function of F1(),F2() : ( P1[g1] & g1 . F3() = F4() . y )  }  by A6, A27, A25, A28;
then A30: g in F . y by A24, A29;
 F . y in rng F by A6, A28, A27, A25, FUNCT_1:def_3;
hence  x in union (rng F) by A23, A30, TARSKI:def_4; ::_thesis: verum
end;
 union (rng F) c=  {  g where g is Function of F1(),F2() : P1[g]  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (rng F) or x in  {  g where g is Function of F1(),F2() : P1[g]  }  )
assume  x in union (rng F) ; ::_thesis: x in  {  g where g is Function of F1(),F2() : P1[g]  }
then consider Y being set  such that
A31: x in Y and
A32: Y in rng F by TARSKI:def_4;
consider X being set  such that
A33: X in dom F and
A34: F . X = Y by A32, FUNCT_1:def_3;
 Y =  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . X )  }  by A6, A33, A34;
then  ex gX being Function of F1(),F2() st
( x = gX & P1[gX] & gX . F3() = F4() . X ) by A31;
hence  x in  {  g where g is Function of F1(),F2() : P1[g]  }  ; ::_thesis: verum
end;
hence  card  {  g where g is Function of F1(),F2() : P1[g]  }  = Sum CardF by A21, A22, XBOOLE_0:def_10; ::_thesis: verum
end;
 for i being Nat st i in dom CardF holds
CardF . i = card  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . i )  }
proof
let i be Nat; ::_thesis: ( i in dom CardF implies CardF . i = card  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . i )  }  )
assume A35: i in dom CardF ; ::_thesis: CardF . i = card  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . i )  }
 F . i =  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . i )  }  by A6, A19, A35;
hence  CardF . i = card  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . i )  }  by A20, A35; ::_thesis: verum
end;
hence  for i being Nat st i in dom CardF holds
CardF . i = card  {  g where g is Function of F1(),F2() : ( P1[g] & g . F3() = F4() . i )  }  ; ::_thesis: verum
end;

theorem Th45: :: STIRL2_1:45
for k, n being Nat holds k * (n block k) = card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }
proof
let k, n be Nat; ::_thesis: k * (n block k) = card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }
now__::_thesis:_k_*_(n_block_k)_=_card__{__f_where_f_is_Function_of_(n_+_1),k_:_(_f_is_onto_&_f_is_"increasing_&_f_"_{(f_._n)}_<>_{n}_)__}_
percases ( k = 0 or k > 0 ) ;
supposeA1: k = 0 ; ::_thesis: k * (n block k) = card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }
set F1 =  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing )  }  ;
set F =  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  ;
A2:  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  c=  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  or x in  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing )  }  )
assume  x in  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  ; ::_thesis: x in  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing )  }
then  ex f being Function of (n + 1),k st
( x = f & f is onto & f is "increasing & f " {(f . n)} <> {n} ) ;
hence  x in  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing )  }  ; ::_thesis: verum
end;
 card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing )  }  = (n + 1) block k ;
then   {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing )  }  is empty by A1, Th31;
hence  k * (n block k) = card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  by A1, A2; ::_thesis: verum
end;
suppose k > 0 ; ::_thesis: k * (n block k) = card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }
then A3: not k is empty ;
set G1 =  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} )  }  ;
defpred S1[ set ] means  ex f being Function of (n + 1),k st
( f = $1 & f is onto & f is "increasing & f " {(f . n)} <> {n} );
 n < n + 1 by NAT_1:13;
then A4: n in n + 1 by NAT_1:44;
 card k,k are_equipotent by CARD_1:def_2;
then consider f being Function such that
A5: f is one-to-one and
A6: dom f = card k and
A7: rng f = k by WELLORD2:def_4;
reconsider f = f as Function of (card k),k by A6, A7, FUNCT_2:1;
A8: ( f is onto & f is one-to-one ) by A5, A7, FUNCT_2:def_3;
consider F being XFinSequence of NAT  such that
A9: dom F = card k and
A10: card  {  g where g is Function of (n + 1),k : S1[g]  }  = Sum F and
A11: for i being Nat st i in dom F holds
F . i = card  {  g where g is Function of (n + 1),k : ( S1[g] & g . n = f . i )  }  from STIRL2_1:sch_6(A8, A3, A4);
A12: for i being Nat st i in dom F holds
F . i = n block k
proof
set F2 =  {  g where g is Function of n,k : ( g is onto & g is "increasing )  }  ;
let i be Nat; ::_thesis: ( i in dom F implies F . i = n block k )
assume A13: i in dom F ; ::_thesis: F . i = n block k
A14: f . i in rng f by A6, A9, A13, FUNCT_1:def_3;
 k is Subset of NAT by Th8;
then reconsider fi = f . i as Element of NAT by A7, A14;
A15: fi < k by A14, NAT_1:44;
set F1 =  {  g where g is Function of (n + 1),k : ( S1[g] & g . n = fi )  }  ;
set F =  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} & g . n = fi )  }  ;
A16:  {  g where g is Function of (n + 1),k : ( S1[g] & g . n = fi )  }  c=  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} & g . n = fi )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  g where g is Function of (n + 1),k : ( S1[g] & g . n = fi )  }  or x in  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} & g . n = fi )  }  )
assume  x in  {  g where g is Function of (n + 1),k : ( S1[g] & g . n = fi )  }  ; ::_thesis: x in  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} & g . n = fi )  }
then  ex g being Function of (n + 1),k st
( x = g & S1[g] & g . n = fi ) ;
hence  x in  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} & g . n = fi )  }  ; ::_thesis: verum
end;
  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} & g . n = fi )  }  c=  {  g where g is Function of (n + 1),k : ( S1[g] & g . n = fi )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} & g . n = fi )  }  or x in  {  g where g is Function of (n + 1),k : ( S1[g] & g . n = fi )  }  )
assume  x in  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} & g . n = fi )  }  ; ::_thesis: x in  {  g where g is Function of (n + 1),k : ( S1[g] & g . n = fi )  }
then  ex g being Function of (n + 1),k st
( x = g & g is onto & g is "increasing & g " {(g . n)} <> {n} & g . n = fi ) ;
hence  x in  {  g where g is Function of (n + 1),k : ( S1[g] & g . n = fi )  }  ; ::_thesis: verum
end;
then   {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} & g . n = fi )  }  =  {  g where g is Function of (n + 1),k : ( S1[g] & g . n = fi )  }  by A16, XBOOLE_0:def_10;
then  card  {  g where g is Function of (n + 1),k : ( S1[g] & g . n = fi )  }  = card  {  g where g is Function of n,k : ( g is onto & g is "increasing )  }  by A15, Th43;
hence  F . i = n block k by A11, A13; ::_thesis: verum
end;
then  for i being Nat st i in dom F holds
F . i <= n block k ;
then A17: Sum F <= (len F) * (n block k) by AFINSQ_2:59;
set G =  {  g where g is Function of (n + 1),k : S1[g]  }  ;
A18:  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} )  }  c=  {  g where g is Function of (n + 1),k : S1[g]  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} )  }  or x in  {  g where g is Function of (n + 1),k : S1[g]  }  )
assume  x in  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} )  }  ; ::_thesis: x in  {  g where g is Function of (n + 1),k : S1[g]  }
then  ex g being Function of (n + 1),k st
( x = g & g is onto & g is "increasing & g " {(g . n)} <> {n} ) ;
hence  x in  {  g where g is Function of (n + 1),k : S1[g]  }  ; ::_thesis: verum
end;
A19:  {  g where g is Function of (n + 1),k : S1[g]  }  c=  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  g where g is Function of (n + 1),k : S1[g]  }  or x in  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} )  }  )
assume  x in  {  g where g is Function of (n + 1),k : S1[g]  }  ; ::_thesis: x in  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} )  }
then  ex g being Function of (n + 1),k st
( x = g & S1[g] ) ;
hence  x in  {  g where g is Function of (n + 1),k : ( g is onto & g is "increasing & g " {(g . n)} <> {n} )  }  ; ::_thesis: verum
end;
 for i being Nat st i in dom F holds
F . i >= n block k by A12;
then A20: Sum F >= (len F) * (n block k) by AFINSQ_2:60;
 card k = k by CARD_1:def_2;
then  Sum F = k * (n block k) by A9, A17, A20, XXREAL_0:1;
hence  k * (n block k) = card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  by A10, A18, A19, XBOOLE_0:def_10; ::_thesis: verum
end;
end;
end;
hence  k * (n block k) = card  {  f where f is Function of (n + 1),k : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  ; ::_thesis: verum
end;

theorem Th46: :: STIRL2_1:46
for n, k being Nat holds (n + 1) block (k + 1) = ((k + 1) * (n block (k + 1))) + (n block k)
proof
let n, k be Nat; ::_thesis: (n + 1) block (k + 1) = ((k + 1) * (n block (k + 1))) + (n block k)
set F =  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing )  }  ;
set F1 =  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  ;
set F2 =  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  ;
A1:  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing )  }  c=  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  \/  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing )  }  or x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  \/  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  )
assume  x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing )  }  ; ::_thesis: x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  \/  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }
then consider f being Function of (n + 1),(k + 1) such that
A2: f = x and
A3: ( f is onto & f is "increasing ) ;
 ( f " {(f . n)} = {n} or f " {(f . n)} <> {n} ) ;
then  ( f in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  or f in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  ) by A3;
hence  x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  \/  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  by A2, XBOOLE_0:def_3; ::_thesis: verum
end;
  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  \/  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  c=  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing )  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  \/  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  or x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing )  }  )
assume  x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  \/  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  ; ::_thesis: x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing )  }
then  ( x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  or x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  ) by XBOOLE_0:def_3;
then  ( ex f being Function of (n + 1),(k + 1) st
( f = x & f is onto & f is "increasing & f " {(f . n)} = {n} ) or ex f being Function of (n + 1),(k + 1) st
( f = x & f is onto & f is "increasing & f " {(f . n)} <> {n} ) ) ;
hence  x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing )  }  ; ::_thesis: verum
end;
then A4:  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  \/  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  =  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing )  }  by A1, XBOOLE_0:def_10;
A5:  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  misses  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }
proof
assume   {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  meets  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  ; ::_thesis: contradiction
then consider x being set  such that
A6: x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  /\  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  by XBOOLE_0:4;
 x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  by A6, XBOOLE_0:def_4;
then A7: ex f being Function of (n + 1),(k + 1) st
( f = x & f is onto & f is "increasing & f " {(f . n)} <> {n} ) ;
 x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  by A6, XBOOLE_0:def_4;
then  ex f being Function of (n + 1),(k + 1) st
( f = x & f is onto & f is "increasing & f " {(f . n)} = {n} ) ;
hence  contradiction by A7; ::_thesis: verum
end;
A8:  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  c= Funcs ((n + 1),(k + 1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  or x in Funcs ((n + 1),(k + 1)) )
assume  x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  ; ::_thesis: x in Funcs ((n + 1),(k + 1))
then  ex f being Function of (n + 1),(k + 1) st
( f = x & f is onto & f is "increasing & f " {(f . n)} <> {n} ) ;
hence  x in Funcs ((n + 1),(k + 1)) by FUNCT_2:8; ::_thesis: verum
end;
A9:  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  c= Funcs ((n + 1),(k + 1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  or x in Funcs ((n + 1),(k + 1)) )
assume  x in  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  ; ::_thesis: x in Funcs ((n + 1),(k + 1))
then  ex f being Function of (n + 1),(k + 1) st
( f = x & f is onto & f is "increasing & f " {(f . n)} = {n} ) ;
hence  x in Funcs ((n + 1),(k + 1)) by FUNCT_2:8; ::_thesis: verum
end;
 Funcs ((n + 1),(k + 1)) is finite by FRAENKEL:6;
then reconsider F1 =  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} = {n} )  }  , F2 =  {  f where f is Function of (n + 1),(k + 1) : ( f is onto & f is "increasing & f " {(f . n)} <> {n} )  }  as finite set by A9, A8;
reconsider k1 = k + 1 as Element of NAT ;
A10: card F2 = k1 * (n block k1) by Th45;
 card F1 = n block k by Th42;
hence  (n + 1) block (k + 1) = ((k + 1) * (n block (k + 1))) + (n block k) by A4, A5, A10, CARD_2:40; ::_thesis: verum
end;

theorem Th47: :: STIRL2_1:47
for n being Nat st n >= 1 holds
n block 2 = (1 / 2) * ((2 |^ n) - 2)
proof
let n be Nat; ::_thesis: ( n >= 1 implies n block 2 = (1 / 2) * ((2 |^ n) - 2) )
defpred S1[ Nat] means $1 block 2 = (1 / 2) * ((2 |^ $1) - 2);
A1: for k being Nat st k >= 1 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 1 & S1[k] implies S1[k + 1] )
assume that
A2: k >= 1 and
A3: S1[k] ; ::_thesis: S1[k + 1]
(k + 1) block 2 = (2 * (k block (1 + 1))) + (k block 1) by Th46
.= (2 * ((1 / 2) * ((2 |^ k) - 2))) + 1 by A2, A3, Th32
.= (1 / 2) * ((2 * (2 |^ k)) - 2)
.= (1 / 2) * ((2 |^ (k + 1)) - 2) by NEWTON:6 ;
hence  S1[k + 1] ; ::_thesis: verum
end;
 2 |^ 1 = 2 by NEWTON:5;
then A4: S1[1] by Th29;
 for k being Nat st k >= 1 holds
S1[k] from NAT_1:sch_8(A4, A1);
hence  ( n >= 1 implies n block 2 = (1 / 2) * ((2 |^ n) - 2) ) ; ::_thesis: verum
end;

theorem Th48: :: STIRL2_1:48
for n being Nat st n >= 2 holds
n block 3 = (1 / 6) * (((3 |^ n) - (3 * (2 |^ n))) + 3)
proof
let n be Nat; ::_thesis: ( n >= 2 implies n block 3 = (1 / 6) * (((3 |^ n) - (3 * (2 |^ n))) + 3) )
defpred S1[ Nat] means $1 block 3 = (1 / 6) * (((3 |^ $1) - (3 * (2 |^ $1))) + 3);
A1: for k being Nat st k >= 2 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 2 & S1[k] implies S1[k + 1] )
assume that
A2: k >= 2 and
A3: S1[k] ; ::_thesis: S1[k + 1]
 k block 2 = (1 / 2) * ((2 |^ k) - 2) by A2, Th47, XXREAL_0:2;
hence (k + 1) block 3 = (3 * (k block (2 + 1))) + ((1 / 2) * ((2 |^ k) - 2)) by Th46
.= (1 / 6) * (((3 * (3 |^ k)) - ((3 * 2) * (2 |^ k))) + 3) by A3
.= (1 / 6) * (((3 |^ (k + 1)) - (3 * (2 * (2 |^ k)))) + 3) by NEWTON:6
.= (1 / 6) * (((3 |^ (k + 1)) - (3 * (2 |^ (k + 1)))) + 3) by NEWTON:6 ;
::_thesis: verum
end;
(1 / 6) * (((3 |^ 2) - (3 * (2 |^ 2))) + 3) = (1 / 6) * (((3 * 3) - (3 * (2 |^ 2))) + 3) by WSIERP_1:1
.= (1 / 6) * (((3 * 3) - (3 * (2 * 2))) + 3) by WSIERP_1:1
.= 2 block 3 by Th29 ;
then A4: S1[2] ;
 for k being Nat st k >= 2 holds
S1[k] from NAT_1:sch_8(A4, A1);
hence  ( n >= 2 implies n block 3 = (1 / 6) * (((3 |^ n) - (3 * (2 |^ n))) + 3) ) ; ::_thesis: verum
end;

Lm3:  for n being Nat holds n |^ 3 = (n * n) * n
proof
let n be Nat; ::_thesis: n |^ 3 = (n * n) * n
reconsider n = n as Element of NAT by ORDINAL1:def_12;
n |^ (2 + 1) = (n |^ 2) * n by NEWTON:6
.= (n * n) * n by WSIERP_1:1 ;
hence  n |^ 3 = (n * n) * n ; ::_thesis: verum
end;

theorem :: STIRL2_1:49
for n being Nat st n >= 3 holds
n block 4 = (1 / 24) * ((((4 |^ n) - (4 * (3 |^ n))) + (6 * (2 |^ n))) - 4)
proof
let n be Nat; ::_thesis: ( n >= 3 implies n block 4 = (1 / 24) * ((((4 |^ n) - (4 * (3 |^ n))) + (6 * (2 |^ n))) - 4) )
defpred S1[ Nat] means $1 block 4 = (1 / 24) * ((((4 |^ $1) - (4 * (3 |^ $1))) + (6 * (2 |^ $1))) - 4);
A1: for k being Nat st k >= 3 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 3 & S1[k] implies S1[k + 1] )
assume that
A2: k >= 3 and
A3: S1[k] ; ::_thesis: S1[k + 1]
 k block 3 = (1 / 6) * (((3 |^ k) - (3 * (2 |^ k))) + 3) by A2, Th48, XXREAL_0:2;
hence (k + 1) block 4 = (4 * (k block (3 + 1))) + ((1 / 6) * (((3 |^ k) - (3 * (2 |^ k))) + 3)) by Th46
.= (1 / 24) * ((((4 * (4 |^ k)) - (4 * (3 * (3 |^ k)))) + (6 * (2 * (2 |^ k)))) - 4) by A3
.= (1 / 24) * ((((4 |^ (k + 1)) - (4 * (3 * (3 |^ k)))) + (6 * (2 * (2 |^ k)))) - 4) by NEWTON:6
.= (1 / 24) * ((((4 |^ (k + 1)) - (4 * (3 |^ (k + 1)))) + (6 * (2 * (2 |^ k)))) - 4) by NEWTON:6
.= (1 / 24) * ((((4 |^ (k + 1)) - (4 * (3 |^ (k + 1)))) + (6 * (2 |^ (k + 1)))) - 4) by NEWTON:6 ;
::_thesis: verum
end;
(1 / 24) * ((((4 |^ 3) - (4 * (3 |^ 3))) + (6 * (2 |^ 3))) - 4) = (1 / 24) * (((((4 * 4) * 4) - (4 * (3 |^ 3))) + (6 * (2 |^ 3))) - 4) by Lm3
.= (1 / 24) * (((64 - (4 * ((3 * 3) * 3))) + (6 * (2 |^ 3))) - 4) by Lm3
.= (1 / 24) * (((64 - (4 * 27)) + (6 * ((2 * 2) * 2))) - 4) by Lm3
.= 3 block 4 by Th29 ;
then A4: S1[3] ;
 for k being Nat st k >= 3 holds
S1[k] from NAT_1:sch_8(A4, A1);
hence  ( n >= 3 implies n block 4 = (1 / 24) * ((((4 |^ n) - (4 * (3 |^ n))) + (6 * (2 |^ n))) - 4) ) ; ::_thesis: verum
end;

theorem Th50: :: STIRL2_1:50
( 3 ! = 6 & 4 ! = 24 )
proof
thus A1: 3 ! = (2 + 1) !
.= 2 * 3 by NEWTON:14, NEWTON:15
.= 6 ; ::_thesis: 4 ! = 24
thus 4 ! = (3 + 1) !
.= 6 * 4 by A1, NEWTON:15
.= 24 ; ::_thesis: verum
end;

theorem Th51: :: STIRL2_1:51
for n being Nat holds
( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 )
proof
let n be Nat; ::_thesis: ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 )
now__::_thesis:_(_n_choose_1_=_n_&_n_choose_2_=_(n_*_(n_-_1))_/_2_&_n_choose_3_=_((n_*_(n_-_1))_*_(n_-_2))_/_6_&_n_choose_4_=_(((n_*_(n_-_1))_*_(n_-_2))_*_(n_-_3))_/_24_)
percases ( n = 0 or n = 1 or n = 2 or n = 3 or n > 3 ) by NAT_1:27;
suppose n = 0 ; ::_thesis: ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 )
hence  ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 ) by NEWTON:def_3; ::_thesis: verum
end;
suppose n = 1 ; ::_thesis: ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 )
hence  ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 ) by NEWTON:21, NEWTON:def_3; ::_thesis: verum
end;
suppose n = 2 ; ::_thesis: ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 )
hence  ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 ) by NEWTON:21, NEWTON:23, NEWTON:def_3; ::_thesis: verum
end;
supposeA1: n = 3 ; ::_thesis: ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 )
then reconsider n1 = n - 1, n2 = n - 2 as Element of NAT by NAT_1:20;
A2: (n2 + 1) ! = (n2 !) * (n2 + 1) by NEWTON:15;
 n2 ! <> 0 by NEWTON:17;
then A3: (n2 !) / (n2 !) = 1 by XCMPLX_1:60;
 (n1 + 1) ! = (n1 !) * n by NEWTON:15;
then n choose 2 = ((n2 !) * ((n - 1) * n)) / ((n2 !) * (2 !)) by A1, A2, NEWTON:def_3
.= (((n2 !) / (n2 !)) * ((n - 1) * n)) / 2 by NEWTON:14, XCMPLX_1:83
.= ((n - 1) * n) / 2 by A3 ;
hence  ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 ) by A1, NEWTON:21, NEWTON:23, NEWTON:def_3; ::_thesis: verum
end;
supposeA4: n > 3 ; ::_thesis: ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 )
then  n >= 3 + 1 by NAT_1:13;
then reconsider n1 = n - 1, n2 = n - 2, n3 = n - 3, n4 = n - 4 as Element of NAT by NAT_1:21, XXREAL_0:2;
A5: (n1 + 1) ! = (n1 !) * n by NEWTON:15;
A6: (n2 + 1) ! = (n2 !) * (n2 + 1) by NEWTON:15;
 n2 ! <> 0 by NEWTON:17;
then A7: (n2 !) / (n2 !) = 1 by XCMPLX_1:60;
 n >= 2 by A4, XXREAL_0:2;
then A8: n choose 2 = ((n2 !) * (n1 * n)) / ((n2 !) * (2 !)) by A5, A6, NEWTON:def_3
.= (((n2 !) / (n2 !)) * (n1 * n)) / 2 by NEWTON:14, XCMPLX_1:83
.= (n * n1) / 2 by A7 ;
 n4 ! <> 0 by NEWTON:17;
then A9: (n4 !) / (n4 !) = 1 by XCMPLX_1:60;
A10: (n4 + 1) ! = (n4 !) * (n4 + 1) by NEWTON:15;
 n3 ! <> 0 by NEWTON:17;
then A11: (n3 !) / (n3 !) = 1 by XCMPLX_1:60;
A12: (n3 + 1) ! = (n3 !) * (n3 + 1) by NEWTON:15;
then A13: n choose 3 = ((n3 !) * ((n2 * n1) * n)) / ((n3 !) * (3 !)) by A4, A5, A6, NEWTON:def_3
.= (((n3 !) / (n3 !)) * ((n2 * n1) * n)) / 6 by Th50, XCMPLX_1:83
.= ((n * n1) * n2) / 6 by A11 ;
 n >= 3 + 1 by A4, NAT_1:13;
then n choose 4 = ((n4 !) * (((n3 * n2) * n1) * n)) / ((n4 !) * (4 !)) by A5, A6, A12, A10, NEWTON:def_3
.= (((n4 !) / (n4 !)) * (((n3 * n2) * n1) * n)) / 24 by Th50, XCMPLX_1:83
.= (((n * n1) * n2) * n3) / 24 by A9 ;
hence  ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 ) by A4, A8, A13, NEWTON:23, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  ( n choose 1 = n & n choose 2 = (n * (n - 1)) / 2 & n choose 3 = ((n * (n - 1)) * (n - 2)) / 6 & n choose 4 = (((n * (n - 1)) * (n - 2)) * (n - 3)) / 24 ) ; ::_thesis: verum
end;

theorem Th52: :: STIRL2_1:52
for n being Nat holds (n + 1) block n = (n + 1) choose 2
proof
let n be Nat; ::_thesis: (n + 1) block n = (n + 1) choose 2
defpred S1[ Nat] means ($1 + 1) block $1 = ($1 + 1) choose 2;
A1: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A2: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
thus ((k + 1) + 1) block (k + 1) = ((k + 1) * ((k + 1) block (k + 1))) + ((k + 1) block k) by Th46
.= ((k + 1) * 1) + ((k + 1) choose 2) by A2, Th26
.= (k + 1) + (((k + 1) * ((k + 1) - 1)) / 2) by Th51
.= (((k + 1) + 1) * (((k + 1) + 1) - 1)) / 2
.= ((k + 1) + 1) choose 2 by Th51 ; ::_thesis: verum
end;
 1 block 0 = 0 by Th31;
then A3: S1[ 0 ] by NEWTON:def_3;
 for k being Nat holds S1[k] from NAT_1:sch_2(A3, A1);
hence  (n + 1) block n = (n + 1) choose 2 ; ::_thesis: verum
end;

theorem :: STIRL2_1:53
for n being Nat holds (n + 2) block n = (3 * ((n + 2) choose 4)) + ((n + 2) choose 3)
proof
let n be Nat; ::_thesis: (n + 2) block n = (3 * ((n + 2) choose 4)) + ((n + 2) choose 3)
defpred S1[ Nat] means ($1 + 2) block $1 = (3 * (($1 + 2) choose 4)) + (($1 + 2) choose 3);
A1: 2 choose 3 = 0 by NEWTON:def_3;
A2: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
set k2 = k + 2;
set k3 = (k + 2) + 1;
A4: (k + 1) * (((k + 1) + 1) block (k + 1)) = (k + 1) * ((k + 2) choose 2) by Th52
.= (k + 1) * (((k + 2) * ((k + 2) - 1)) / 2) by Th51
.= ((k + 2) * ((k + 2) - 1)) * (((k + 1) * 12) / 24) ;
(k + 2) block k = (3 * (((((k + 2) * ((k + 2) - 1)) * ((k + 2) - 2)) * ((k + 2) - 3)) / 24)) + ((k + 2) choose 3) by A3, Th51
.= (3 * (((((k + 2) * ((k + 2) - 1)) * ((k + 2) - 2)) * ((k + 2) - 3)) / 24)) + ((((k + 2) * ((k + 2) - 1)) * ((k + 2) - 2)) / 6) by Th51
.= ((k + 2) * ((k + 2) - 1)) * ((((3 * ((k + 2) - 2)) * ((k + 2) - 3)) / 24) + ((4 * ((k + 2) - 2)) / 24)) ;
then ((k + 2) + 1) block (k + 1) = (((k + 2) * (k + 1)) * (((k + 1) * 12) / 24)) + (((k + 2) * (k + 1)) * ((((3 * k) * ((k + 2) - 3)) / 24) + ((4 * k) / 24))) by A4, Th46
.= (3 * ((((((k + 2) + 1) * (((k + 2) + 1) - 1)) * (((k + 2) + 1) - 2)) * (((k + 2) + 1) - 3)) / 24)) + (((((k + 2) + 1) * (k + 2)) * (k + 1)) / 6)
.= (3 * (((k + 2) + 1) choose 4)) + (((((k + 2) + 1) * (((k + 2) + 1) - 1)) * (((k + 2) + 1) - 2)) / 6) by Th51
.= (3 * (((k + 2) + 1) choose 4)) + (((k + 2) + 1) choose 3) by Th51 ;
hence  S1[k + 1] ; ::_thesis: verum
end;
 2 choose 4 = 0 by NEWTON:def_3;
then A5: S1[ 0 ] by A1, Th31;
 for k being Nat holds S1[k] from NAT_1:sch_2(A5, A2);
hence  (n + 2) block n = (3 * ((n + 2) choose 4)) + ((n + 2) choose 3) ; ::_thesis: verum
end;

theorem Th54: :: STIRL2_1:54
for F being Function
for y being set holds
( rng (F | ((dom F) \ (F " {y}))) = (rng F) \ {y} & ( for x being set st x <> y holds
(F | ((dom F) \ (F " {y}))) " {x} = F " {x} ) )
proof
let F be Function; ::_thesis: for y being set holds
( rng (F | ((dom F) \ (F " {y}))) = (rng F) \ {y} & ( for x being set st x <> y holds
(F | ((dom F) \ (F " {y}))) " {x} = F " {x} ) )
let y be set ; ::_thesis: ( rng (F | ((dom F) \ (F " {y}))) = (rng F) \ {y} & ( for x being set st x <> y holds
(F | ((dom F) \ (F " {y}))) " {x} = F " {x} ) )
set D = (dom F) \ (F " {y});
A1: rng (F | ((dom F) \ (F " {y}))) c= (rng F) \ {y}
proof
let Fx be set ; :: according to TARSKI:def_3 ::_thesis: ( not Fx in rng (F | ((dom F) \ (F " {y}))) or Fx in (rng F) \ {y} )
assume  Fx in rng (F | ((dom F) \ (F " {y}))) ; ::_thesis: Fx in (rng F) \ {y}
then consider x being set  such that
A2: x in dom (F | ((dom F) \ (F " {y}))) and
A3: Fx = (F | ((dom F) \ (F " {y}))) . x by FUNCT_1:def_3;
A4: x in (dom F) /\ ((dom F) \ (F " {y})) by A2, RELAT_1:61;
then  x in ((dom F) /\ (dom F)) \ (F " {y}) by XBOOLE_1:49;
then  not x in F " {y} by XBOOLE_0:def_5;
then  not F . x in {y} by A4, FUNCT_1:def_7;
then A5: not Fx in {y} by A2, A3, FUNCT_1:47;
 F . x in rng F by A4, FUNCT_1:def_3;
then  Fx in rng F by A2, A3, FUNCT_1:47;
hence  Fx in (rng F) \ {y} by A5, XBOOLE_0:def_5; ::_thesis: verum
end;
 (rng F) \ {y} c= rng (F | ((dom F) \ (F " {y})))
proof
let Fx be set ; :: according to TARSKI:def_3 ::_thesis: ( not Fx in (rng F) \ {y} or Fx in rng (F | ((dom F) \ (F " {y}))) )
assume A6: Fx in (rng F) \ {y} ; ::_thesis: Fx in rng (F | ((dom F) \ (F " {y})))
consider x being set  such that
A7: x in dom F and
A8: F . x = Fx by A6, FUNCT_1:def_3;
 not Fx in {y} by A6, XBOOLE_0:def_5;
then  not x in F " {y} by A8, FUNCT_1:def_7;
then  x in (dom F) \ (F " {y}) by A7, XBOOLE_0:def_5;
then  x in (dom F) /\ ((dom F) \ (F " {y})) by XBOOLE_0:def_4;
then A9: x in dom (F | ((dom F) \ (F " {y}))) by RELAT_1:61;
then  (F | ((dom F) \ (F " {y}))) . x in rng (F | ((dom F) \ (F " {y}))) by FUNCT_1:def_3;
hence  Fx in rng (F | ((dom F) \ (F " {y}))) by A8, A9, FUNCT_1:47; ::_thesis: verum
end;
hence  rng (F | ((dom F) \ (F " {y}))) = (rng F) \ {y} by A1, XBOOLE_0:def_10; ::_thesis: for x being set st x <> y holds
(F | ((dom F) \ (F " {y}))) " {x} = F " {x}
let x be set ; ::_thesis: ( x <> y implies (F | ((dom F) \ (F " {y}))) " {x} = F " {x} )
assume A10: x <> y ; ::_thesis: (F | ((dom F) \ (F " {y}))) " {x} = F " {x}
now__::_thesis:_for_z_being_set_st_z_in_F_"_{x}_holds_
z_in_(dom_F)_\_(F_"_{y})
let z be set ; ::_thesis: ( z in F " {x} implies z in (dom F) \ (F " {y}) )
assume A11: z in F " {x} ; ::_thesis: z in (dom F) \ (F " {y})
 F . z in {x} by A11, FUNCT_1:def_7;
then  F . z <> y by A10, TARSKI:def_1;
then  not F . z in {y} by TARSKI:def_1;
then A12: not z in F " {y} by FUNCT_1:def_7;
 z in dom F by A11, FUNCT_1:def_7;
hence  z in (dom F) \ (F " {y}) by A12, XBOOLE_0:def_5; ::_thesis: verum
end;
then  F " {x} c= (dom F) \ (F " {y}) by TARSKI:def_3;
hence  (F | ((dom F) \ (F " {y}))) " {x} = F " {x} by FUNCT_2:98; ::_thesis: verum
end;

theorem Th55: :: STIRL2_1:55
for k being Nat
for X, x being set st card X = k + 1 & x in X holds
card (X \ {x}) = k
proof
let k be Nat; ::_thesis: for X, x being set st card X = k + 1 & x in X holds
card (X \ {x}) = k
let X, x be set ; ::_thesis: ( card X = k + 1 & x in X implies card (X \ {x}) = k )
assume that
A1: card X = k + 1 and
A2: x in X ; ::_thesis: card (X \ {x}) = k
reconsider X9 = X as finite set by A1;
set Xx = X9 \ {x};
 {x} c= X by A2, ZFMISC_1:31;
then  {x} /\ X = {x} by XBOOLE_1:28;
then  (X9 \ {x}) \/ {x} = X by XBOOLE_1:51;
then A3: (card {x}) + (card (X9 \ {x})) = k + 1 by A1, CARD_2:40, XBOOLE_1:79;
 card {x} = 1 by CARD_1:30;
hence  card (X \ {x}) = k by A3; ::_thesis: verum
end;

scheme :: STIRL2_1:sch 7
Sch9{ P1[ set ], P2[ set , Function] } :
 for F being Function st rng F is finite holds
P1[F]
provided
A1: P1[ {} ] and
A2: for F being Function st ( for x being set st x in rng F & P2[x,F] holds
P1[F | ((dom F) \ (F " {x}))] ) holds
P1[F]
proof
defpred S1[ Nat] means  for F being Function st card (rng F) = $1 holds
P1[F];
A3: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A4: S1[n] ; ::_thesis: S1[n + 1]
let F be Function; ::_thesis: ( card (rng F) = n + 1 implies P1[F] )
assume A5: card (rng F) = n + 1 ; ::_thesis: P1[F]
 for x being set st x in rng F & P2[x,F] holds
P1[F | ((dom F) \ (F " {x}))]
proof
let x be set ; ::_thesis: ( x in rng F & P2[x,F] implies P1[F | ((dom F) \ (F " {x}))] )
assume that
A6: x in rng F and
 P2[x,F] ; ::_thesis: P1[F | ((dom F) \ (F " {x}))]
set D = (dom F) \ (F " {x});
 card ((rng F) \ {x}) = n by A5, A6, Th55;
then  card (rng (F | ((dom F) \ (F " {x})))) = n by Th54;
hence  P1[F | ((dom F) \ (F " {x}))] by A4; ::_thesis: verum
end;
hence  P1[F] by A2; ::_thesis: verum
end;
let F be Function; ::_thesis: ( rng F is finite implies P1[F] )
assume  rng F is finite ; ::_thesis: P1[F]
then consider n being Nat such that
A7: rng F,n are_equipotent by CARD_1:43;
A8: card (rng F) = n by A7, CARD_1:def_2;
A9: S1[ 0 ]
proof
let F be Function; ::_thesis: ( card (rng F) = 0 implies P1[F] )
assume  card (rng F) = 0 ; ::_thesis: P1[F]
then  rng F = {} ;
hence  P1[F] by A1, RELAT_1:41; ::_thesis: verum
end;
 for k being Nat holds S1[k] from NAT_1:sch_2(A9, A3);
hence  P1[F] by A8; ::_thesis: verum
end;

theorem Th56: :: STIRL2_1:56
for N being Subset of NAT st N is finite holds
ex k being Nat st
for n being Nat st n in N holds
n <= k
proof
let N be Subset of NAT; ::_thesis: ( N is finite implies ex k being Nat st
for n being Nat st n in N holds
n <= k )
assume  N is finite ; ::_thesis: ex k being Nat st
for n being Nat st n in N holds
n <= k
then consider n being Nat such that
A1: N,n are_equipotent by CARD_1:43;
consider F being Function such that
 F is one-to-one and
A2: dom F = n and
A3: rng F = N by A1, WELLORD2:def_4;
reconsider F = F as XFinSequence by A2, AFINSQ_1:5;
reconsider F = F as XFinSequence of NAT by A3, RELAT_1:def_19;
reconsider k = Sum F as Element of NAT by ORDINAL1:def_12;
take  k ; ::_thesis: for n being Nat st n in N holds
n <= k
let n be Nat; ::_thesis: ( n in N implies n <= k )
assume A4: n in N ; ::_thesis: n <= k
 F <> 0 by A3, A4;
then A5: len F > 0 ;
 ex x being set st
( x in dom F & F . x = n ) by A3, A4, FUNCT_1:def_3;
hence  n <= k by A5, AFINSQ_2:61; ::_thesis: verum
end;

theorem Th57: :: STIRL2_1:57
for X, Y, x, y being set st ( Y is empty implies X is empty ) & not x in X holds
for F being Function of X,Y ex G being Function of (X \/ {x}),(Y \/ {y}) st
( G | X = F & G . x = y )
proof
let X, Y, x, y be set ; ::_thesis: ( ( Y is empty implies X is empty ) & not x in X implies for F being Function of X,Y ex G being Function of (X \/ {x}),(Y \/ {y}) st
( G | X = F & G . x = y ) )
assume that
A1: ( Y is empty implies X is empty ) and
A2: not x in X ; ::_thesis: for F being Function of X,Y ex G being Function of (X \/ {x}),(Y \/ {y}) st
( G | X = F & G . x = y )
set Y1 = Y \/ {y};
set X1 = X \/ {x};
deffunc H1( set ) ->  set = y;
let F be Function of X,Y; ::_thesis: ex G being Function of (X \/ {x}),(Y \/ {y}) st
( G | X = F & G . x = y )
 y in {y} by TARSKI:def_1;
then A3: for x9 being set st x9 in (X \/ {x}) \ X holds
H1(x9) in Y \/ {y} by XBOOLE_0:def_3;
A4: ( X c= X \/ {x} & Y c= Y \/ {y} ) by XBOOLE_1:7;
consider G being Function of (X \/ {x}),(Y \/ {y}) such that
A5: ( G | X = F & ( for x9 being set st x9 in (X \/ {x}) \ X holds
G . x9 = H1(x9) ) ) from STIRL2_1:sch_2(A3, A4, A1);
 x in {x} by TARSKI:def_1;
then  x in X \/ {x} by XBOOLE_0:def_3;
then  x in (X \/ {x}) \ X by A2, XBOOLE_0:def_5;
then  G . x = y by A5;
hence  ex G being Function of (X \/ {x}),(Y \/ {y}) st
( G | X = F & G . x = y ) by A5; ::_thesis: verum
end;

theorem Th58: :: STIRL2_1:58
for X, Y, x, y being set st ( Y is empty implies X is empty ) holds
for F being Function of X,Y
for G being Function of (X \/ {x}),(Y \/ {y}) st G | X = F & G . x = y holds
( ( F is onto implies G is onto ) & ( not y in Y & F is one-to-one implies G is one-to-one ) )
proof
let X, Y, x, y be set ; ::_thesis: ( ( Y is empty implies X is empty ) implies for F being Function of X,Y
for G being Function of (X \/ {x}),(Y \/ {y}) st G | X = F & G . x = y holds
( ( F is onto implies G is onto ) & ( not y in Y & F is one-to-one implies G is one-to-one ) ) )
assume A1: ( Y is empty implies X is empty ) ; ::_thesis: for F being Function of X,Y
for G being Function of (X \/ {x}),(Y \/ {y}) st G | X = F & G . x = y holds
( ( F is onto implies G is onto ) & ( not y in Y & F is one-to-one implies G is one-to-one ) )
let F be Function of X,Y; ::_thesis: for G being Function of (X \/ {x}),(Y \/ {y}) st G | X = F & G . x = y holds
( ( F is onto implies G is onto ) & ( not y in Y & F is one-to-one implies G is one-to-one ) )
let G be Function of (X \/ {x}),(Y \/ {y}); ::_thesis: ( G | X = F & G . x = y implies ( ( F is onto implies G is onto ) & ( not y in Y & F is one-to-one implies G is one-to-one ) ) )
assume that
A2: G | X = F and
A3: G . x = y ; ::_thesis: ( ( F is onto implies G is onto ) & ( not y in Y & F is one-to-one implies G is one-to-one ) )
thus  ( F is onto implies G is onto ) ::_thesis: ( not y in Y & F is one-to-one implies G is one-to-one )
proof
assume A4: F is onto ; ::_thesis: G is onto
 Y \/ {y} c= rng G
proof
let Fx be set ; :: according to TARSKI:def_3 ::_thesis: ( not Fx in Y \/ {y} or Fx in rng G )
assume A5: Fx in Y \/ {y} ; ::_thesis: Fx in rng G
now__::_thesis:_Fx_in_rng_G
percases ( Fx in Y or Fx in {y} ) by A5, XBOOLE_0:def_3;
suppose Fx in Y ; ::_thesis: Fx in rng G
then  Fx in rng F by A4, FUNCT_2:def_3;
then consider x9 being set  such that
A6: x9 in dom F and
A7: F . x9 = Fx by FUNCT_1:def_3;
A8: x9 in X by A6;
A9: dom G = X \/ {x} by FUNCT_2:def_1;
A10: X c= X \/ {x} by XBOOLE_1:7;
 G . x9 = F . x9 by A2, A6, FUNCT_1:47;
hence  Fx in rng G by A7, A8, A10, A9, FUNCT_1:def_3; ::_thesis: verum
end;
supposeA11: Fx in {y} ; ::_thesis: Fx in rng G
A12: dom G = X \/ {x} by FUNCT_2:def_1;
A13: {x} c= X \/ {x} by XBOOLE_1:7;
A14: x in {x} by TARSKI:def_1;
 Fx = y by A11, TARSKI:def_1;
hence  Fx in rng G by A3, A12, A14, A13, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
hence  Fx in rng G ; ::_thesis: verum
end;
then  Y \/ {y} = rng G by XBOOLE_0:def_10;
hence  G is onto by FUNCT_2:def_3; ::_thesis: verum
end;
thus  ( not y in Y & F is one-to-one implies G is one-to-one ) ::_thesis: verum
proof
assume that
A15: not y in Y and
A16: F is one-to-one ; ::_thesis: G is one-to-one
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom G or not x2 in dom G or not G . x1 = G . x2 or x1 = x2 )
assume that
A17: x1 in dom G and
A18: x2 in dom G and
A19: G . x1 = G . x2 ; ::_thesis: x1 = x2
A20: for x9 being set st x9 in X holds
( x9 in dom F & F . x9 = G . x9 & G . x9 <> y )
proof
let x9 be set ; ::_thesis: ( x9 in X implies ( x9 in dom F & F . x9 = G . x9 & G . x9 <> y ) )
assume A21: x9 in X ; ::_thesis: ( x9 in dom F & F . x9 = G . x9 & G . x9 <> y )
A22: x9 in dom F by A1, A21, FUNCT_2:def_1;
then A23: F . x9 in rng F by FUNCT_1:def_3;
 F . x9 = G . x9 by A2, A22, FUNCT_1:47;
hence  ( x9 in dom F & F . x9 = G . x9 & G . x9 <> y ) by A15, A21, A23, FUNCT_2:def_1; ::_thesis: verum
end;
now__::_thesis:_x1_=_x2
percases ( ( x1 in X & x2 in X ) or ( x1 in X & x2 in {x} ) or ( x1 in {x} & x2 in X ) or ( x1 in {x} & x2 in {x} ) ) by A17, A18, XBOOLE_0:def_3;
supposeA24: ( x1 in X & x2 in X ) ; ::_thesis: x1 = x2
then A25: F . x1 = G . x1 by A20;
A26: x2 in dom F by A20, A24;
A27: F . x2 = G . x2 by A20, A24;
 x1 in dom F by A20, A24;
hence  x1 = x2 by A16, A19, A26, A25, A27, FUNCT_1:def_4; ::_thesis: verum
end;
supposeA28: ( x1 in X & x2 in {x} ) ; ::_thesis: x1 = x2
then  G . x2 = y by A3, TARSKI:def_1;
hence  x1 = x2 by A19, A20, A28; ::_thesis: verum
end;
supposeA29: ( x1 in {x} & x2 in X ) ; ::_thesis: x1 = x2
then  G . x1 = y by A3, TARSKI:def_1;
hence  x1 = x2 by A19, A20, A29; ::_thesis: verum
end;
supposeA30: ( x1 in {x} & x2 in {x} ) ; ::_thesis: x1 = x2
then  x = x1 by TARSKI:def_1;
hence  x1 = x2 by A30, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence  x1 = x2 ; ::_thesis: verum
end;
end;

theorem Th59: :: STIRL2_1:59
for N being finite Subset of NAT ex Order being Function of N,(card N) st
( Order is bijective & ( for n, k being Nat st n in dom Order & k in dom Order & n < k holds
Order . n < Order . k ) )
proof
defpred S1[ Nat] means  for N being finite Subset of NAT st card N = $1 holds
ex F being Function of N,(card N) st
( F is bijective & ( for n, k being Nat st n in dom F & k in dom F & n < k holds
F . n < F . k ) );
A1: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A2: S1[k] ; ::_thesis: S1[k + 1]
let N be finite Subset of NAT; ::_thesis: ( card N = k + 1 implies ex F being Function of N,(card N) st
( F is bijective & ( for n, k being Nat st n in dom F & k in dom F & n < k holds
F . n < F . k ) ) )
assume A3: card N = k + 1 ; ::_thesis: ex F being Function of N,(card N) st
( F is bijective & ( for n, k being Nat st n in dom F & k in dom F & n < k holds
F . n < F . k ) )
defpred S2[ set ] means $1 in N;
 ex x being set st x in N by A3, CARD_1:27, XBOOLE_0:def_1;
then A4: ex n being Nat st S2[n] ;
consider m9 being Nat such that
A5: for n being Nat st n in N holds
n <= m9 by Th56;
A6: for n being Nat st S2[n] holds
n <= m9 by A5;
consider m being Nat such that
A7: ( S2[m] & ( for n being Nat st S2[n] holds
n <= m ) ) from NAT_1:sch_6(A6, A4);
set Nm = N \ {m};
consider F being Function of (N \ {m}),(card (N \ {m})) such that
A8: F is bijective and
A9: for n, k being Nat st n in dom F & k in dom F & n < k holds
F . n < F . k by A2, A3, A7, Th55;
A10: card (N \ {m}) = k by A3, A7, Th55;
then A11: (card (N \ {m})) \/ {k} = card N by A3, AFINSQ_1:2;
A12: ( card (N \ {m}) is empty implies N \ {m} is empty ) ;
 m in {m} by TARSKI:def_1;
then  not m in N \ {m} by XBOOLE_0:def_5;
then consider G being Function of ((N \ {m}) \/ {m}),((card (N \ {m})) \/ {k}) such that
A13: G | (N \ {m}) = F and
A14: G . m = k by A12, Th57;
 N = (N \ {m}) \/ {m} by A7, ZFMISC_1:116;
then reconsider G9 = G as Function of N,(card N) by A3, A10, AFINSQ_1:2;
take  G9 ; ::_thesis: ( G9 is bijective & ( for n, k being Nat st n in dom G9 & k in dom G9 & n < k holds
G9 . n < G9 . k ) )
 not k in card (N \ {m}) by A10;
then  ( G is one-to-one & G is onto ) by A8, A12, A13, A14, Th58;
hence  G9 is bijective by A11; ::_thesis: for n, k being Nat st n in dom G9 & k in dom G9 & n < k holds
G9 . n < G9 . k
thus  for n, k being Nat st n in dom G9 & k in dom G9 & n < k holds
G9 . n < G9 . k ::_thesis: verum
proof
A15: for i being Nat st i in N \ {m} holds
( i < m & G9 . i < k )
proof
let i be Nat; ::_thesis: ( i in N \ {m} implies ( i < m & G9 . i < k ) )
assume A16: i in N \ {m} ; ::_thesis: ( i < m & G9 . i < k )
 not i in {m} by A16, XBOOLE_0:def_5;
then A17: i <> m by TARSKI:def_1;
 S2[i] by A16, XBOOLE_0:def_5;
then A18: i <= m by A7;
 i in dom F by A16, FUNCT_2:def_1;
then A19: F . i = G9 . i by A13, FUNCT_1:47;
 card (N \ {m}) = k by A3, A7, Th55;
hence  ( i < m & G9 . i < k ) by A16, A19, A18, A17, NAT_1:44, XXREAL_0:1; ::_thesis: verum
end;
let i, j be Nat; ::_thesis: ( i in dom G9 & j in dom G9 & i < j implies G9 . i < G9 . j )
assume that
A20: i in dom G9 and
A21: j in dom G9 and
A22: i < j ; ::_thesis: G9 . i < G9 . j
A23: dom G9 = (N \ {m}) \/ {m} by FUNCT_2:def_1;
now__::_thesis:_G9_._i_<_G9_._j
percases ( ( i in N \ {m} & j in N \ {m} ) or ( i in N \ {m} & j in {m} ) or ( i in {m} & j in N \ {m} ) or ( i in {m} & j in {m} ) ) by A20, A21, A23, XBOOLE_0:def_3;
supposeA24: ( i in N \ {m} & j in N \ {m} ) ; ::_thesis: G9 . i < G9 . j
then A25: j in dom F by FUNCT_2:def_1;
then A26: F . j = G9 . j by A13, FUNCT_1:47;
A27: i in dom F by A24, FUNCT_2:def_1;
then  F . i = G9 . i by A13, FUNCT_1:47;
hence  G9 . i < G9 . j by A9, A22, A27, A25, A26; ::_thesis: verum
end;
supposeA28: ( i in N \ {m} & j in {m} ) ; ::_thesis: G9 . i < G9 . j
then  G9 . i < k by A15;
hence  G9 . i < G9 . j by A14, A28, TARSKI:def_1; ::_thesis: verum
end;
supposeA29: ( i in {m} & j in N \ {m} ) ; ::_thesis: G9 . i < G9 . j
then  i = m by TARSKI:def_1;
hence  G9 . i < G9 . j by A22, A15, A29; ::_thesis: verum
end;
supposeA30: ( i in {m} & j in {m} ) ; ::_thesis: G9 . i < G9 . j
then  i = m by TARSKI:def_1;
hence  G9 . i < G9 . j by A22, A30, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence  G9 . i < G9 . j ; ::_thesis: verum
end;
end;
A31: S1[ 0 ]
proof
set P = {} ;
A32: rng {} = {} ;
A33: dom {} = {} ;
let N be finite Subset of NAT; ::_thesis: ( card N = 0 implies ex F being Function of N,(card N) st
( F is bijective & ( for n, k being Nat st n in dom F & k in dom F & n < k holds
F . n < F . k ) ) )
assume A34: card N = 0 ; ::_thesis: ex F being Function of N,(card N) st
( F is bijective & ( for n, k being Nat st n in dom F & k in dom F & n < k holds
F . n < F . k ) )
 N is empty by A34;
then reconsider P = {} as Function of N,(card N) by A34, A32, A33, FUNCT_2:1;
take  P ; ::_thesis: ( P is bijective & ( for n, k being Nat st n in dom P & k in dom P & n < k holds
P . n < P . k ) )
 rng P = {} ;
then  ( P is one-to-one & P is onto ) by A34, FUNCT_2:def_3;
hence  P is bijective ; ::_thesis: for n, k being Nat st n in dom P & k in dom P & n < k holds
P . n < P . k
thus  for n, k being Nat st n in dom P & k in dom P & n < k holds
P . n < P . k ; ::_thesis: verum
end;
 for k being Nat holds S1[k] from NAT_1:sch_2(A31, A1);
hence  for N being finite Subset of NAT ex Order being Function of N,(card N) st
( Order is bijective & ( for n, k being Nat st n in dom Order & k in dom Order & n < k holds
Order . n < Order . k ) ) ; ::_thesis: verum
end;

Lm4:  for X being finite set
for x being set st x in X holds
card (X \ {x}) < card X
proof
let X be finite set ; ::_thesis: for x being set st x in X holds
card (X \ {x}) < card X
let x be set ; ::_thesis: ( x in X implies card (X \ {x}) < card X )
assume A1: x in X ; ::_thesis: card (X \ {x}) < card X
 card X > 0 by A1;
then reconsider c1 = (card X) - 1 as Element of NAT by NAT_1:20;
 c1 < c1 + 1 by NAT_1:13;
hence  card (X \ {x}) < card X by A1, Th55; ::_thesis: verum
end;

theorem Th60: :: STIRL2_1:60
for X, Y being finite set
for F being Function of X,Y st card X = card Y holds
( F is onto iff F is one-to-one )
proof
let X, Y be finite set ; ::_thesis: for F being Function of X,Y st card X = card Y holds
( F is onto iff F is one-to-one )
let F be Function of X,Y; ::_thesis: ( card X = card Y implies ( F is onto iff F is one-to-one ) )
assume A1: card X = card Y ; ::_thesis: ( F is onto iff F is one-to-one )
thus  ( F is onto implies F is one-to-one ) ::_thesis: ( F is one-to-one implies F is onto )
proof
assume A2: F is onto ; ::_thesis: F is one-to-one
assume  not F is one-to-one ; ::_thesis: contradiction
then consider x1, x2 being set  such that
A3: x1 in dom F and
A4: x2 in dom F and
A5: F . x1 = F . x2 and
A6: x1 <> x2 by FUNCT_1:def_4;
reconsider Xx = X \ {x1} as finite set ;
 Y c= F .: Xx
proof
let Fy be set ; :: according to TARSKI:def_3 ::_thesis: ( not Fy in Y or Fy in F .: Xx )
assume  Fy in Y ; ::_thesis: Fy in F .: Xx
then  Fy in rng F by A2, FUNCT_2:def_3;
then consider y being set  such that
A7: y in dom F and
A8: F . y = Fy by FUNCT_1:def_3;
now__::_thesis:_Fy_in_F_.:_Xx
percases ( y = x1 or y <> x1 ) ;
supposeA9: y = x1 ; ::_thesis: Fy in F .: Xx
 x2 in Xx by A4, A6, ZFMISC_1:56;
hence  Fy in F .: Xx by A4, A5, A8, A9, FUNCT_1:def_6; ::_thesis: verum
end;
suppose y <> x1 ; ::_thesis: Fy in F .: Xx
then  y in Xx by A7, ZFMISC_1:56;
hence  Fy in F .: Xx by A7, A8, FUNCT_1:def_6; ::_thesis: verum
end;
end;
end;
hence  Fy in F .: Xx ; ::_thesis: verum
end;
then A10: card Y c= card Xx by CARD_1:66;
 card Xx < card X by A3, Lm4;
hence  contradiction by A1, A10, NAT_1:39; ::_thesis: verum
end;
thus  ( F is one-to-one implies F is onto ) ::_thesis: verum
proof
assume  F is one-to-one ; ::_thesis: F is onto
then  dom F,F .: (dom F) are_equipotent by CARD_1:33;
then A11: card (dom F) = card (F .: (dom F)) by CARD_1:5;
assume  not F is onto ; ::_thesis: contradiction
then  not rng F = Y by FUNCT_2:def_3;
then  not Y c= rng F by XBOOLE_0:def_10;
then consider y being set  such that
A12: y in Y and
A13: not y in rng F by TARSKI:def_3;
A14: card (rng F) <= card (Y \ {y}) by A13, NAT_1:43, ZFMISC_1:34;
A15: F .: (dom F) = rng F by RELAT_1:113;
 card (Y \ {y}) < card Y by A12, Lm4;
hence  contradiction by A1, A12, A14, A11, A15, FUNCT_2:def_1; ::_thesis: verum
end;
end;

Lm5:  for n being Element of NAT
for N being finite Subset of NAT
for F being Function of N,(card N) st n in N & F is bijective & ( for n, k being Nat st n in dom F & k in dom F & n < k holds
F . n < F . k ) holds
ex P being Permutation of N st
for k being Nat st k in N holds
( ( k < n implies P . k = (F ") . ((F . k) + 1) ) & ( k = n implies P . k = (F ") . 0 ) & ( k > n implies P . k = k ) )
proof
let n be Element of NAT ; ::_thesis: for N being finite Subset of NAT
for F being Function of N,(card N) st n in N & F is bijective & ( for n, k being Nat st n in dom F & k in dom F & n < k holds
F . n < F . k ) holds
ex P being Permutation of N st
for k being Nat st k in N holds
( ( k < n implies P . k = (F ") . ((F . k) + 1) ) & ( k = n implies P . k = (F ") . 0 ) & ( k > n implies P . k = k ) )
let N be finite Subset of NAT; ::_thesis: for F being Function of N,(card N) st n in N & F is bijective & ( for n, k being Nat st n in dom F & k in dom F & n < k holds
F . n < F . k ) holds
ex P being Permutation of N st
for k being Nat st k in N holds
( ( k < n implies P . k = (F ") . ((F . k) + 1) ) & ( k = n implies P . k = (F ") . 0 ) & ( k > n implies P . k = k ) )
let F be Function of N,(card N); ::_thesis: ( n in N & F is bijective & ( for n, k being Nat st n in dom F & k in dom F & n < k holds
F . n < F . k ) implies ex P being Permutation of N st
for k being Nat st k in N holds
( ( k < n implies P . k = (F ") . ((F . k) + 1) ) & ( k = n implies P . k = (F ") . 0 ) & ( k > n implies P . k = k ) ) )
assume that
A1: n in N and
A2: F is bijective and
A3: for n, k being Nat st n in dom F & k in dom F & n < k holds
F . n < F . k ; ::_thesis: ex P being Permutation of N st
for k being Nat st k in N holds
( ( k < n implies P . k = (F ") . ((F . k) + 1) ) & ( k = n implies P . k = (F ") . 0 ) & ( k > n implies P . k = k ) )
 rng F = card N by A2, FUNCT_2:def_3;
then reconsider F9 = F " as Function of (card N),N by A2, FUNCT_2:25;
defpred S1[ set , set ] means  for k being Nat st k = $1 holds
( ( k < n implies $2 = (F ") . ((F . k) + 1) ) & ( k = n implies $2 = (F ") . 0 ) & ( k > n implies $2 = k ) );
A4: dom F = N by A1, FUNCT_2:def_1;
A5: dom F9 = card N by A1, FUNCT_2:def_1;
A6: for x being set st x in N holds
ex y being set st
( y in N & S1[x,y] )
proof
let x9 be set ; ::_thesis: ( x9 in N implies ex y being set st
( y in N & S1[x9,y] ) )
assume A7: x9 in N ; ::_thesis: ex y being set st
( y in N & S1[x9,y] )
reconsider x = x9 as Element of NAT by A7;
now__::_thesis:_ex_y_being_set_st_
(_y_in_N_&_S1[x9,y]_)
percases ( x < n or x = n or x > n ) by XXREAL_0:1;
supposeA8: x < n ; ::_thesis: ex y being set st
( y in N & S1[x9,y] )
then  F . x < F . n by A1, A3, A4, A7;
then A9: (F . x) + 1 <= F . n by NAT_1:13;
 F . n < card N by A1, NAT_1:44;
then  (F . x) + 1 < card N by A9, XXREAL_0:2;
then  (F . x) + 1 in dom F9 by A5, NAT_1:44;
then A10: F9 . ((F . x) + 1) in rng F9 by FUNCT_1:def_3;
set FF = (F ") . ((F . x) + 1);
 S1[x9,(F ") . ((F . x) + 1)] by A8;
hence  ex y being set st
( y in N & S1[x9,y] ) by A10; ::_thesis: verum
end;
supposeA11: x = n ; ::_thesis: ex y being set st
( y in N & S1[x9,y] )
 0 in dom F9 by A1, A5, NAT_1:44;
then A12: F9 . 0 in rng F9 by FUNCT_1:def_3;
 S1[x9,(F ") . 0] by A11;
hence  ex y being set st
( y in N & S1[x9,y] ) by A12; ::_thesis: verum
end;
suppose x > n ; ::_thesis: ex y being set st
( y in N & S1[x9,y] )
then  S1[x,x] ;
hence  ex y being set st
( y in N & S1[x9,y] ) by A7; ::_thesis: verum
end;
end;
end;
hence  ex y being set st
( y in N & S1[x9,y] ) ; ::_thesis: verum
end;
consider P being Function of N,N such that
A13: for x being set st x in N holds
S1[x,P . x] from FUNCT_2:sch_1(A6);
 N c= rng P
proof
let Px9 be set ; :: according to TARSKI:def_3 ::_thesis: ( not Px9 in N or Px9 in rng P )
assume A14: Px9 in N ; ::_thesis: Px9 in rng P
reconsider Px = Px9 as Element of NAT by A14;
now__::_thesis:_Px9_in_rng_P
percases ( Px <= n or Px > n ) ;
supposeA15: Px <= n ; ::_thesis: Px9 in rng P
 rng F9 = N by A2, A4, FUNCT_1:33;
then consider x being set  such that
A16: x in dom F9 and
A17: F9 . x = Px by A14, FUNCT_1:def_3;
 card N is Subset of NAT by Th8;
then reconsider x = x as Element of NAT by A5, A16;
now__::_thesis:_Px9_in_rng_P
percases ( x = 0 or x > 0 ) ;
supposeA18: x = 0 ; ::_thesis: Px9 in rng P
A19: dom P = N by A1, FUNCT_2:def_1;
 P . n = (F ") . 0 by A1, A13;
hence  Px9 in rng P by A1, A17, A18, A19, FUNCT_1:def_3; ::_thesis: verum
end;
suppose x > 0 ; ::_thesis: Px9 in rng P
then reconsider x1 = x - 1 as Element of NAT by NAT_1:20;
A20: x1 < x1 + 1 by NAT_1:13;
 x < card N by A16, NAT_1:44;
then  x1 < card N by A20, XXREAL_0:2;
then A21: x1 in card N by NAT_1:44;
 card N = rng F by A2, FUNCT_2:def_3;
then consider y being set  such that
A22: y in dom F and
A23: F . y = x1 by A21, FUNCT_1:def_3;
reconsider y = y as Element of NAT by A4, A22;
A24: y in dom P by A4, A22, FUNCT_2:def_1;
A25: y < n
proof
assume  y >= n ; ::_thesis: contradiction
then  ( y > n or y = n ) by XXREAL_0:1;
then A26: x1 >= F . n by A1, A3, A4, A22, A23;
 x in rng F by A2, A16, FUNCT_1:33;
then A27: F . Px = x by A2, A17, FUNCT_1:32;
 ( Px < n or Px = n ) by A15, XXREAL_0:1;
then A28: F . Px <= F . n by A1, A3, A4, A14;
 x1 + 1 > x1 by NAT_1:13;
hence  contradiction by A26, A27, A28, XXREAL_0:2; ::_thesis: verum
end;
 (F ") . ((F . y) + 1) = Px by A17, A23;
then  P . y = Px by A13, A22, A25;
hence  Px9 in rng P by A24, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
hence  Px9 in rng P ; ::_thesis: verum
end;
supposeA29: Px > n ; ::_thesis: Px9 in rng P
A30: Px in dom P by A14, FUNCT_2:def_1;
 Px = P . Px by A13, A14, A29;
hence  Px9 in rng P by A30, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
hence  Px9 in rng P ; ::_thesis: verum
end;
then  N = rng P by XBOOLE_0:def_10;
then A31: P is onto by FUNCT_2:def_3;
 card N = card N ;
then  ( P is onto & P is one-to-one ) by A31, Th60;
then reconsider P = P as Permutation of N ;
take  P ; ::_thesis: for k being Nat st k in N holds
( ( k < n implies P . k = (F ") . ((F . k) + 1) ) & ( k = n implies P . k = (F ") . 0 ) & ( k > n implies P . k = k ) )
thus  for k being Nat st k in N holds
( ( k < n implies P . k = (F ") . ((F . k) + 1) ) & ( k = n implies P . k = (F ") . 0 ) & ( k > n implies P . k = k ) ) by A13; ::_thesis: verum
end;

theorem Th61: :: STIRL2_1:61
for F, G being Function
for y being set st y in rng (G * F) & G is one-to-one holds
ex x being set st
( x in dom G & x in rng F & G " {y} = {x} & F " {x} = (G * F) " {y} )
proof
let F, G be Function; ::_thesis: for y being set st y in rng (G * F) & G is one-to-one holds
ex x being set st
( x in dom G & x in rng F & G " {y} = {x} & F " {x} = (G * F) " {y} )
let y be set ; ::_thesis: ( y in rng (G * F) & G is one-to-one implies ex x being set st
( x in dom G & x in rng F & G " {y} = {x} & F " {x} = (G * F) " {y} ) )
assume that
A1: y in rng (G * F) and
A2: G is one-to-one ; ::_thesis: ex x being set st
( x in dom G & x in rng F & G " {y} = {x} & F " {x} = (G * F) " {y} )
consider x being set  such that
A3: x in dom (G * F) and
A4: (G * F) . x = y by A1, FUNCT_1:def_3;
A5: F . x in dom G by A3, FUNCT_1:11;
A6: G . (F . x) = y by A3, A4, FUNCT_1:12;
then  G . (F . x) in {y} by TARSKI:def_1;
then A7: F . x in G " {y} by A5, FUNCT_1:def_7;
A8: F " {(F . x)} c= (G * F) " {y}
proof
let d be set ; :: according to TARSKI:def_3 ::_thesis: ( not d in F " {(F . x)} or d in (G * F) " {y} )
assume A9: d in F " {(F . x)} ; ::_thesis: d in (G * F) " {y}
A10: d in dom F by A9, FUNCT_1:def_7;
 F . d in {(F . x)} by A9, FUNCT_1:def_7;
then A11: F . d = F . x by TARSKI:def_1;
then  G . (F . d) in {y} by A6, TARSKI:def_1;
then A12: (G * F) . d in {y} by A10, FUNCT_1:13;
 d in dom (G * F) by A5, A10, A11, FUNCT_1:11;
hence  d in (G * F) " {y} by A12, FUNCT_1:def_7; ::_thesis: verum
end;
 y in rng G by A1, FUNCT_1:14;
then consider Fx being set  such that
A13: G " {y} = {Fx} by A2, FUNCT_1:74;
 x in dom F by A3, FUNCT_1:11;
then A14: F . x in rng F by FUNCT_1:def_3;
A15: F . x in dom G by A3, FUNCT_1:11;
 (G * F) " {y} c= F " {(F . x)}
proof
let d be set ; :: according to TARSKI:def_3 ::_thesis: ( not d in (G * F) " {y} or d in F " {(F . x)} )
assume A16: d in (G * F) " {y} ; ::_thesis: d in F " {(F . x)}
A17: d in dom (G * F) by A16, FUNCT_1:def_7;
then A18: d in dom F by FUNCT_1:11;
 (G * F) . d in {y} by A16, FUNCT_1:def_7;
then A19: G . (F . d) in {y} by A17, FUNCT_1:12;
A20: F . d in dom G by A17, FUNCT_1:11;
 F . x = Fx by A13, A7, TARSKI:def_1;
then  F . d in {(F . x)} by A13, A20, A19, FUNCT_1:def_7;
hence  d in F " {(F . x)} by A18, FUNCT_1:def_7; ::_thesis: verum
end;
then A21: F " {(F . x)} = (G * F) " {y} by A8, XBOOLE_0:def_10;
 G " {y} = {(F . x)} by A13, A7, TARSKI:def_1;
hence  ex x being set st
( x in dom G & x in rng F & G " {y} = {x} & F " {x} = (G * F) " {y} ) by A15, A14, A21; ::_thesis: verum
end;

definition
let Ne, Ke be Subset of NAT;
let f be Function of Ne,Ke;
attrf is "increasing  means :Def3: :: STIRL2_1:def 3
 for l, m being Nat st l in rng f & m in rng f & l < m holds
min* (f " {l}) < min* (f " {m});
end;

:: deftheorem Def3 defines "increasing STIRL2_1:def_3_:_
 for Ne, Ke being Subset of NAT
for f being Function of Ne,Ke holds
( f is "increasing iff for l, m being Nat st l in rng f & m in rng f & l < m holds
min* (f " {l}) < min* (f " {m}) );

theorem Th62: :: STIRL2_1:62
for Ne, Ke being Subset of NAT
for F being Function of Ne,Ke st F is "increasing holds
min* (rng F) = F . (min* (dom F))
proof
let Ne, Ke be Subset of NAT; ::_thesis: for F being Function of Ne,Ke st F is "increasing holds
min* (rng F) = F . (min* (dom F))
let F be Function of Ne,Ke; ::_thesis: ( F is "increasing implies min* (rng F) = F . (min* (dom F)) )
assume A1: F is "increasing ; ::_thesis: min* (rng F) = F . (min* (dom F))
now__::_thesis:_min*_(rng_F)_=_F_._(min*_(dom_F))
percases ( rng F is empty or not rng F is empty ) ;
supposeA2: rng F is empty ; ::_thesis: min* (rng F) = F . (min* (dom F))
then A3: min* (rng F) = 0 by NAT_1:def_1;
 dom F = {} by A2, RELAT_1:42;
hence  min* (rng F) = F . (min* (dom F)) by A3, FUNCT_1:def_2; ::_thesis: verum
end;
supposeA4: not rng F is empty ; ::_thesis: min* (rng F) = F . (min* (dom F))
then reconsider rngF = rng F, Ke = Ke as non empty Subset of NAT by XBOOLE_1:1;
 not Ke is empty ;
then reconsider domF = dom F as non empty Subset of NAT by A4, FUNCT_2:def_1, RELAT_1:42;
set md = min* domF;
set mr = min* rngF;
 min* rngF = F . (min* domF)
proof
A5: min* domF in dom F by NAT_1:def_1;
then  F . (min* domF) in rngF by FUNCT_1:def_3;
then A6: min* rngF <= F . (min* domF) by NAT_1:def_1;
assume  min* rngF <> F . (min* domF) ; ::_thesis: contradiction
then A7: min* rngF < F . (min* domF) by A6, XXREAL_0:1;
A8: min* domF in domF by NAT_1:def_1;
A9: min* domF in dom F by NAT_1:def_1;
 min* rngF in rngF by NAT_1:def_1;
then consider x being set  such that
A10: x in dom F and
A11: F . x = min* rngF by FUNCT_1:def_3;
A12: F . (min* domF) in {(F . (min* domF))} by TARSKI:def_1;
 F . x in {(min* rngF)} by A11, TARSKI:def_1;
then reconsider Fmr = F " {(min* rngF)}, Fmd = F " {(F . (min* domF))} as non empty Subset of NAT by A10, A12, A9, FUNCT_1:def_7, XBOOLE_1:1;
A13: min* rngF in rngF by NAT_1:def_1;
 min* Fmr in Fmr by NAT_1:def_1;
then  min* Fmr in domF by FUNCT_1:def_7;
then A14: min* Fmr >= min* domF by NAT_1:def_1;
 F . (min* domF) in {(F . (min* domF))} by TARSKI:def_1;
then  min* domF in Fmd by A8, FUNCT_1:def_7;
then A15: min* domF >= min* Fmd by NAT_1:def_1;
 F . (min* domF) in rng F by A5, FUNCT_1:def_3;
then  min* (F " {(min* rngF)}) < min* (F " {(F . (min* domF))}) by A1, A7, A13, Def3;
hence  contradiction by A14, A15, XXREAL_0:2; ::_thesis: verum
end;
hence  min* (rng F) = F . (min* (dom F)) ; ::_thesis: verum
end;
end;
end;
hence  min* (rng F) = F . (min* (dom F)) ; ::_thesis: verum
end;

theorem :: STIRL2_1:63
for Ne, Ke being Subset of NAT
for F being Function of Ne,Ke st rng F is finite holds
ex I being Function of Ne,Ke ex P being Permutation of (rng F) st
( F = P * I & rng F = rng I & I is "increasing )
proof
let Ne, Ke be Subset of NAT; ::_thesis: for F being Function of Ne,Ke st rng F is finite holds
ex I being Function of Ne,Ke ex P being Permutation of (rng F) st
( F = P * I & rng F = rng I & I is "increasing )
defpred S1[ set , Function] means $1 = $2 . (min* (dom $2));
defpred S2[ set ] means  for Ne, Ke being Subset of NAT
for F being Function of Ne,Ke st F = $1 & rng F is finite holds
ex P being Permutation of (rng F) ex G being Function of Ne,Ke st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) );
A1: S2[ {} ]
proof
let Ne, Me be Subset of NAT; ::_thesis: for F being Function of Ne,Me st F = {} & rng F is finite holds
ex P being Permutation of (rng F) ex G being Function of Ne,Me st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) )
let F be Function of Ne,Me; ::_thesis: ( F = {} & rng F is finite implies ex P being Permutation of (rng F) ex G being Function of Ne,Me st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) ) )
assume that
A2: F = {} and
 rng F is finite ; ::_thesis: ex P being Permutation of (rng F) ex G being Function of Ne,Me st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) )
reconsider R = rng F as empty set by A2;
set P = {} ;
A3: rng {} = {} ;
 dom {} = {} ;
then reconsider P = {} as Function of R,R by A3, FUNCT_2:1;
 rng R = {} ;
then  ( P is one-to-one & P is onto ) by FUNCT_2:def_3;
then reconsider P = P as Permutation of (rng F) ;
take  P ; ::_thesis: ex G being Function of Ne,Me st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) )
take  F ; ::_thesis: ( F = P * F & rng F = rng F & ( for i, j being Nat st i in rng F & j in rng F & i < j holds
min* (F " {i}) < min* (F " {j}) ) )
 rng F = R ;
then  F = {} ;
hence  ( F = P * F & rng F = rng F & ( for i, j being Nat st i in rng F & j in rng F & i < j holds
min* (F " {i}) < min* (F " {j}) ) ) ; ::_thesis: verum
end;
A4: for F being Function st ( for x being set st x in rng F & S1[x,F] holds
S2[F | ((dom F) \ (F " {x}))] ) holds
S2[F]
proof
let F9 be Function; ::_thesis: ( ( for x being set st x in rng F9 & S1[x,F9] holds
S2[F9 | ((dom F9) \ (F9 " {x}))] ) implies S2[F9] )
assume A5: for x being set st x in rng F9 & S1[x,F9] holds
S2[F9 | ((dom F9) \ (F9 " {x}))] ; ::_thesis: S2[F9]
let N, K be Subset of NAT; ::_thesis: for F being Function of N,K st F = F9 & rng F is finite holds
ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) )
let F be Function of N,K; ::_thesis: ( F = F9 & rng F is finite implies ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) ) )
assume that
A6: F = F9 and
A7: rng F is finite ; ::_thesis: ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) )
now__::_thesis:_ex_P_being_Permutation_of_(rng_F)_ex_G_being_Function_of_N,K_st_
(_F_=_P_*_G_&_rng_F_=_rng_G_&_(_for_i,_j_being_Nat_st_i_in_rng_G_&_j_in_rng_G_&_i_<_j_holds_
min*_(G_"_{i})_<_min*_(G_"_{j})_)_)
percases ( rng F9 is empty or not rng F9 is empty ) ;
suppose rng F9 is empty ; ::_thesis: ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) )
then  F9 = {} ;
hence  ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) ) by A1, A6; ::_thesis: verum
end;
supposeA8: not rng F9 is empty ; ::_thesis: ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) )
then reconsider domF = dom F as non empty Subset of NAT by A6, RELAT_1:42, XBOOLE_1:1;
reconsider K = K as non empty Subset of NAT by A6, A8;
set m = min* domF;
set D = (dom F) \ (F " {(F . (min* domF))});
 min* domF in domF by NAT_1:def_1;
then A9: F . (min* domF) in rng F by FUNCT_1:def_3;
now__::_thesis:_ex_P_being_Permutation_of_(rng_F)_ex_G_being_Function_of_N,K_st_
(_F_=_P_*_G_&_rng_F_=_rng_G_&_(_for_i,_j_being_Nat_st_i_in_rng_G_&_j_in_rng_G_&_i_<_j_holds_
min*_(G_"_{i})_<_min*_(G_"_{j})_)_)
percases ( rng (F | ((dom F) \ (F " {(F . (min* domF))}))) is empty or not rng (F | ((dom F) \ (F " {(F . (min* domF))}))) is empty ) ;
suppose rng (F | ((dom F) \ (F " {(F . (min* domF))}))) is empty ; ::_thesis: ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) )
then  (rng F) \ {(F . (min* domF))} = {} by Th54;
then A10: rng F = {(F . (min* domF))} by A9, ZFMISC_1:58;
A11: for i, j being Nat st i in rng F & j in rng F & i < j holds
min* (F " {i}) < min* (F " {j})
proof
let i, j be Nat; ::_thesis: ( i in rng F & j in rng F & i < j implies min* (F " {i}) < min* (F " {j}) )
assume that
A12: i in rng F and
A13: j in rng F and
A14: i < j ; ::_thesis: min* (F " {i}) < min* (F " {j})
 i = F . (min* domF) by A10, A12, TARSKI:def_1;
hence  min* (F " {i}) < min* (F " {j}) by A10, A13, A14, TARSKI:def_1; ::_thesis: verum
end;
set P = id (rng F);
 rng (id (rng F)) = rng F ;
then  ( id (rng F) is one-to-one & id (rng F) is onto ) by FUNCT_2:def_3;
then reconsider P = id (rng F) as Permutation of (rng F) ;
 F is Function of (dom F),(rng F) by FUNCT_2:1;
then  P * F = F by FUNCT_2:17;
hence  ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) ) by A11; ::_thesis: verum
end;
supposeA15: not rng (F | ((dom F) \ (F " {(F . (min* domF))}))) is empty ; ::_thesis: ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) )
 rng (F | ((dom F) \ (F " {(F . (min* domF))}))) c= rng F by RELAT_1:70;
then reconsider rFD = rng (F | ((dom F) \ (F " {(F . (min* domF))}))) as non empty finite Subset of NAT by A7, A15, XBOOLE_1:1;
deffunc H1( set ) ->  set = F . (min* domF);
reconsider dFD = dom (F | ((dom F) \ (F " {(F . (min* domF))}))) as Subset of NAT by XBOOLE_1:1;
reconsider FD = F | ((dom F) \ (F " {(F . (min* domF))})) as Function of dFD,rFD by FUNCT_2:1;
A16: ( rFD is empty implies dFD is empty ) ;
reconsider rF = rng F as non empty finite Subset of NAT by A7, A9, XBOOLE_1:1;
A17: ( dFD c= N & rFD c= K ) ;
consider P being Permutation of (rng FD), G being Function of dFD,rFD such that
A18: FD = P * G and
A19: rng FD = rng G and
A20: for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) by A5, A6, A9;
A21: for x being set st x in N \ dFD holds
H1(x) in K by A9;
consider G2 being Function of N,K such that
A22: ( G2 | dFD = G & ( for x being set st x in N \ dFD holds
G2 . x = H1(x) ) ) from STIRL2_1:sch_2(A21, A17, A16);
A23: rng G2 c= rng F
proof
let Gx be set ; :: according to TARSKI:def_3 ::_thesis: ( not Gx in rng G2 or Gx in rng F )
assume  Gx in rng G2 ; ::_thesis: Gx in rng F
then consider x being set  such that
A24: x in dom G2 and
A25: G2 . x = Gx by FUNCT_1:def_3;
 dom G2 = N by FUNCT_2:def_1;
then  ((dom G2) /\ dFD) \/ (N \ dFD) = dom G2 by XBOOLE_1:51;
then  ( ( dom G = (dom G2) /\ dFD & x in (dom G2) /\ dFD ) or x in N \ dFD ) by A22, A24, RELAT_1:61, XBOOLE_0:def_3;
then  ( ( G . x in rng FD & G . x = G2 . x & rng FD = (rng F) \ {(F . (min* domF))} ) or ( G2 . x = F . (min* domF) & min* domF in domF ) ) by A19, A22, Th54, FUNCT_1:47, FUNCT_1:def_3, NAT_1:def_1;
hence  Gx in rng F by A25, FUNCT_1:def_3, XBOOLE_0:def_5; ::_thesis: verum
end;
A26: rng FD = (rng F) \ {(F . (min* domF))} by Th54;
 dom FD = (dom F) /\ ((dom F) \ (F " {(F . (min* domF))})) by RELAT_1:61;
then A27: dFD = (dom F) \ (F " {(F . (min* domF))}) by XBOOLE_1:28, XBOOLE_1:36;
A28: F " {(F . (min* domF))} c= G2 " {(F . (min* domF))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F " {(F . (min* domF))} or x in G2 " {(F . (min* domF))} )
assume A29: x in F " {(F . (min* domF))} ; ::_thesis: x in G2 " {(F . (min* domF))}
 not x in (dom F) \ (F " {(F . (min* domF))}) by A29, XBOOLE_0:def_5;
then  x in N \ dFD by A27, A29, XBOOLE_0:def_5;
then  G2 . x = F . (min* domF) by A22;
then A30: G2 . x in {(F . (min* domF))} by TARSKI:def_1;
 x in N by A29;
then  x in dom G2 by FUNCT_2:def_1;
hence  x in G2 " {(F . (min* domF))} by A30, FUNCT_1:def_7; ::_thesis: verum
end;
 rng F c= rng G2
proof
 rng FD = (rng F) \ {(F . (min* domF))} by Th54;
then A31: rng F = (rng FD) \/ {(F . (min* domF))} by A9, ZFMISC_1:116;
let Fx be set ; :: according to TARSKI:def_3 ::_thesis: ( not Fx in rng F or Fx in rng G2 )
assume A32: Fx in rng F ; ::_thesis: Fx in rng G2
now__::_thesis:_Fx_in_rng_G2
percases ( Fx in rng FD or Fx in {(F . (min* domF))} ) by A32, A31, XBOOLE_0:def_3;
supposeA33: Fx in rng FD ; ::_thesis: Fx in rng G2
 rng (G2 | dFD) c= rng G2 by RELAT_1:70;
hence  Fx in rng G2 by A19, A22, A33; ::_thesis: verum
end;
supposeA34: Fx in {(F . (min* domF))} ; ::_thesis: Fx in rng G2
A35: min* domF in dom F by NAT_1:def_1;
then  min* domF in N ;
then  min* domF in dom G2 by FUNCT_2:def_1;
then A36: G2 . (min* domF) in rng G2 by FUNCT_1:def_3;
 F . (min* domF) in {(F . (min* domF))} by TARSKI:def_1;
then  min* domF in F " {(F . (min* domF))} by A35, FUNCT_1:def_7;
then  not min* domF in (dom F) \ (F " {(F . (min* domF))}) by XBOOLE_0:def_5;
then A37: min* domF in N \ dFD by A27, A35, XBOOLE_0:def_5;
 Fx = F . (min* domF) by A34, TARSKI:def_1;
hence  Fx in rng G2 by A22, A37, A36; ::_thesis: verum
end;
end;
end;
hence  Fx in rng G2 ; ::_thesis: verum
end;
then A38: rng G2 = rng F by A23, XBOOLE_0:def_10;
 not F . (min* domF) in (rng F) \ {(F . (min* domF))} by ZFMISC_1:56;
then  not F . (min* domF) in rng FD by Th54;
then consider P2 being Function of ((rng FD) \/ {(F . (min* domF))}),((rng FD) \/ {(F . (min* domF))}) such that
A39: P2 | (rng FD) = P and
A40: P2 . (F . (min* domF)) = F . (min* domF) by Th57;
 not F . (min* domF) in (rng F) \ {(F . (min* domF))} by ZFMISC_1:56;
then A41: ( P2 is one-to-one & P2 is onto ) by A39, A40, A26, Th58;
 (rng FD) \/ {(F . (min* domF))} = rng F by A9, A26, ZFMISC_1:116;
then reconsider P2 = P2 as Permutation of (rng F) by A41;
consider Orde being Function of rF,(card rF) such that
A42: Orde is bijective and
A43: for n, k being Nat st n in dom Orde & k in dom Orde & n < k holds
Orde . n < Orde . k by Th59;
 rng Orde = card rF by A42, FUNCT_2:def_3;
then reconsider Orde9 = Orde " as Function of (card rF),rF by A42, FUNCT_2:25;
consider P1 being Permutation of rF such that
A44: for k being Nat st k in rF holds
( ( k < F . (min* domF) implies P1 . k = (Orde ") . ((Orde . k) + 1) ) & ( k = F . (min* domF) implies P1 . k = (Orde ") . 0 ) & ( k > F . (min* domF) implies P1 . k = k ) ) by A9, A42, A43, Lm5;
 dom G2 = N by FUNCT_2:def_1;
then A45: G2 is Function of N,rF by A38, FUNCT_2:1;
reconsider P21 = P2 * (P1 ") as Function of rF,rF ;
reconsider P21 = P21 as Permutation of rF ;
 dom P1 = rF by FUNCT_2:def_1;
then A46: (P1 ") * P1 = id rF by FUNCT_1:39;
 rng (P1 * G2) c= K by XBOOLE_1:1;
then reconsider PG = P1 * G2 as Function of N,K by A45, FUNCT_2:6;
 dom G2 = N by FUNCT_2:def_1;
then  G2 is Function of N,rF by A38, FUNCT_2:1;
then  (id rF) * G2 = G2 by FUNCT_2:17;
then A47: (P1 ") * (P1 * G2) = G2 by A46, RELAT_1:36;
 G2 " {(F . (min* domF))} c= F " {(F . (min* domF))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in G2 " {(F . (min* domF))} or x in F " {(F . (min* domF))} )
assume A48: x in G2 " {(F . (min* domF))} ; ::_thesis: x in F " {(F . (min* domF))}
A49: x in N \ dFD
proof
assume  not x in N \ dFD ; ::_thesis: contradiction
then  ( ( x in dom G2 & dom G2 = N & not x in N ) or x in dFD ) by A48, XBOOLE_0:def_5;
then A50: x in dom G by FUNCT_2:def_1;
then A51: G . x in rng G by FUNCT_1:def_3;
A52: rng FD = (rng F) \ {(F . (min* domF))} by Th54;
 G . x = G2 . x by A22, A50, FUNCT_1:47;
then  not G2 . x in {(F . (min* domF))} by A51, A52, XBOOLE_0:def_5;
hence  contradiction by A48, FUNCT_1:def_7; ::_thesis: verum
end;
then A53: not x in (dom F) \ (F " {(F . (min* domF))}) by A27, XBOOLE_0:def_5;
 dom F = N by A9, FUNCT_2:def_1;
then  x in dom F by A49, XBOOLE_0:def_5;
hence  x in F " {(F . (min* domF))} by A53, XBOOLE_0:def_5; ::_thesis: verum
end;
then A54: G2 " {(F . (min* domF))} = F " {(F . (min* domF))} by A28, XBOOLE_0:def_10;
A55: for n being Nat st n in rng FD holds
G " {n} = G2 " {n}
proof
 not K is empty ;
then  dom F = N by FUNCT_2:def_1;
then A56: dFD = (dom G2) \ (G2 " {(F . (min* domF))}) by A27, A54, FUNCT_2:def_1;
A57: rng FD = (rng F) \ {(F . (min* domF))} by Th54;
let n be Nat; ::_thesis: ( n in rng FD implies G " {n} = G2 " {n} )
assume  n in rng FD ; ::_thesis: G " {n} = G2 " {n}
then  not n in {(F . (min* domF))} by A57, XBOOLE_0:def_5;
then  n <> F . (min* domF) by TARSKI:def_1;
hence  G " {n} = G2 " {n} by A22, A56, Th54; ::_thesis: verum
end;
A58: for i, j being Nat st i in rng PG & j in rng PG & i < j holds
min* (PG " {i}) < min* (PG " {j})
proof
A59: for l being Nat st l in rF & l < F . (min* domF) holds
( (Orde . l) + 1 in rng Orde & (Orde . l) + 1 is Element of NAT & (Orde . l) + 1 <= Orde . (F . (min* domF)) & Orde . (F . (min* domF)) is Element of NAT & dom Orde = rF )
proof
A60: Orde . (F . (min* domF)) < card rF by NAT_1:44;
let l be Nat; ::_thesis: ( l in rF & l < F . (min* domF) implies ( (Orde . l) + 1 in rng Orde & (Orde . l) + 1 is Element of NAT & (Orde . l) + 1 <= Orde . (F . (min* domF)) & Orde . (F . (min* domF)) is Element of NAT & dom Orde = rF ) )
assume that
A61: l in rF and
A62: l < F . (min* domF) ; ::_thesis: ( (Orde . l) + 1 in rng Orde & (Orde . l) + 1 is Element of NAT & (Orde . l) + 1 <= Orde . (F . (min* domF)) & Orde . (F . (min* domF)) is Element of NAT & dom Orde = rF )
 dom Orde = rF by FUNCT_2:def_1;
then A63: Orde . l < Orde . (F . (min* domF)) by A9, A43, A61, A62;
then  (Orde . l) + 1 <= Orde . (F . (min* domF)) by NAT_1:13;
then A64: (Orde . l) + 1 < card rF by A60, XXREAL_0:2;
A65: card rF is Subset of NAT by Th8;
A66: Orde . (F . (min* domF)) in card rF ;
 card rF = rng Orde by A42, FUNCT_2:def_3;
hence  ( (Orde . l) + 1 in rng Orde & (Orde . l) + 1 is Element of NAT & (Orde . l) + 1 <= Orde . (F . (min* domF)) & Orde . (F . (min* domF)) is Element of NAT & dom Orde = rF ) by A63, A66, A65, A64, FUNCT_2:def_1, NAT_1:13, NAT_1:44; ::_thesis: verum
end;
A67: for n, k being Nat st n in dom Orde & k in dom Orde & Orde . n < Orde . k holds
n < k
proof
let n, k be Nat; ::_thesis: ( n in dom Orde & k in dom Orde & Orde . n < Orde . k implies n < k )
assume that
A68: n in dom Orde and
A69: k in dom Orde and
A70: Orde . n < Orde . k ; ::_thesis: n < k
assume  n >= k ; ::_thesis: contradiction
then  n > k by A70, XXREAL_0:1;
hence  contradiction by A43, A68, A69, A70; ::_thesis: verum
end;
A71: for n, k being Nat st n in rng Orde & k in rng Orde & n < k holds
Orde9 . n < Orde9 . k
proof
let n, k be Nat; ::_thesis: ( n in rng Orde & k in rng Orde & n < k implies Orde9 . n < Orde9 . k )
assume that
A72: n in rng Orde and
A73: k in rng Orde ; ::_thesis: ( not n < k or Orde9 . n < Orde9 . k )
A74: n = Orde . (Orde9 . n) by A42, A72, FUNCT_1:35;
A75: dom Orde = rng Orde9 by A42, FUNCT_1:33;
A76: k = Orde . (Orde9 . k) by A42, A73, FUNCT_1:35;
assume A77: n < k ; ::_thesis: Orde9 . n < Orde9 . k
A78: rng Orde = dom Orde9 by A42, FUNCT_1:33;
then A79: Orde9 . n in dom Orde by A72, A75, FUNCT_1:def_3;
 Orde9 . k in dom Orde by A73, A78, A75, FUNCT_1:def_3;
hence  Orde9 . n < Orde9 . k by A67, A77, A74, A76, A79; ::_thesis: verum
end;
 rng FD = (rng F) \ {(F . (min* domF))} by Th54;
then A80: (rng FD) \/ {(F . (min* domF))} = rng G2 by A9, A38, ZFMISC_1:116;
let i, j be Nat; ::_thesis: ( i in rng PG & j in rng PG & i < j implies min* (PG " {i}) < min* (PG " {j}) )
assume that
A81: i in rng PG and
A82: j in rng PG and
A83: i < j ; ::_thesis: min* (PG " {i}) < min* (PG " {j})
consider i1 being set  such that
A84: i1 in dom P1 and
A85: i1 in rng G2 and
A86: P1 " {i} = {i1} and
A87: G2 " {i1} = PG " {i} by A81, Th61;
consider j1 being set  such that
A88: j1 in dom P1 and
A89: j1 in rng G2 and
A90: P1 " {j} = {j1} and
A91: G2 " {j1} = PG " {j} by A82, Th61;
 dom P1 = rF by FUNCT_2:def_1;
then reconsider i1 = i1, j1 = j1 as Element of NAT by A84, A88;
A92: i1 in P1 " {i} by A86, TARSKI:def_1;
then  P1 . i1 in {i} by FUNCT_1:def_7;
then A93: P1 . i1 = i by TARSKI:def_1;
A94: j1 in P1 " {j} by A90, TARSKI:def_1;
then A95: P1 . j1 in {j} by FUNCT_1:def_7;
then A96: P1 . j1 = j by TARSKI:def_1;
A97: dom Orde = rF by FUNCT_2:def_1;
now__::_thesis:_min*_(PG_"_{i})_<_min*_(PG_"_{j})
percases ( ( i1 < F . (min* domF) & j1 < F . (min* domF) ) or ( i1 = F . (min* domF) & j1 <> F . (min* domF) ) or ( i1 < F . (min* domF) & j1 = F . (min* domF) ) or ( i1 = F . (min* domF) & j1 = F . (min* domF) ) or ( i1 > F . (min* domF) & j1 > F . (min* domF) ) or ( i1 > F . (min* domF) & j1 = F . (min* domF) ) or ( i1 < F . (min* domF) & j1 > F . (min* domF) ) or ( i1 > F . (min* domF) & j1 < F . (min* domF) ) ) by XXREAL_0:1;
supposeA98: ( i1 < F . (min* domF) & j1 < F . (min* domF) ) ; ::_thesis: min* (PG " {i}) < min* (PG " {j})
A99: ( i1 in rng FD or i1 in {(F . (min* domF))} ) by A85, A80, XBOOLE_0:def_3;
then A100: G " {i1} = PG " {i} by A55, A87, A98, TARSKI:def_1;
A101: ( j1 in rng FD or j1 in {(F . (min* domF))} ) by A89, A80, XBOOLE_0:def_3;
 i1 < j1
proof
assume  i1 >= j1 ; ::_thesis: contradiction
then  i1 > j1 by A83, A93, A96, XXREAL_0:1;
then  Orde . i1 > Orde . j1 by A43, A92, A94, A97;
then A102: (Orde . i1) + 1 > (Orde . j1) + 1 by XREAL_1:8;
A103: (Orde . i1) + 1 in rng Orde by A84, A59, A98;
A104: Orde9 . ((Orde . j1) + 1) = j by A44, A94, A96, A98;
A105: (Orde . j1) + 1 in rng Orde by A88, A59, A98;
 Orde9 . ((Orde . i1) + 1) = i by A44, A92, A93, A98;
hence  contradiction by A83, A71, A102, A103, A105, A104; ::_thesis: verum
end;
then  min* (G " {i1}) < min* (G " {j1}) by A19, A20, A98, A99, A101, TARSKI:def_1;
hence  min* (PG " {i}) < min* (PG " {j}) by A55, A91, A98, A101, A100, TARSKI:def_1; ::_thesis: verum
end;
supposeA106: ( i1 = F . (min* domF) & j1 <> F . (min* domF) ) ; ::_thesis: min* (PG " {i}) < min* (PG " {j})
consider x being set  such that
A107: x in dom G2 and
A108: G2 . x = j1 by A89, FUNCT_1:def_3;
 G2 . x in {j1} by A108, TARSKI:def_1;
then  PG " {j} is non empty Subset of NAT by A91, A107, FUNCT_1:def_7, XBOOLE_1:1;
then A109: min* (PG " {j}) in PG " {j} by NAT_1:def_1;
 ( j1 in rng FD or j1 in {(F . (min* domF))} ) by A89, A80, XBOOLE_0:def_3;
then A110: G " {j1} = PG " {j} by A55, A91, A106, TARSKI:def_1;
then  min* (PG " {j}) in dom F by A27, A109, XBOOLE_0:def_5;
then A111: min* domF <= min* (PG " {j}) by NAT_1:def_1;
A112: min* domF in domF by NAT_1:def_1;
 F . (min* domF) in {(F . (min* domF))} by TARSKI:def_1;
then A113: min* domF in F " {(F . (min* domF))} by A112, FUNCT_1:def_7;
then  min* domF <> min* (PG " {j}) by A27, A110, A109, XBOOLE_0:def_5;
then A114: min* domF < min* (PG " {j}) by A111, XXREAL_0:1;
 PG " {i} is Subset of NAT by XBOOLE_1:1;
then  min* (PG " {i}) <= min* domF by A28, A87, A106, A113, NAT_1:def_1;
hence  min* (PG " {i}) < min* (PG " {j}) by A114, XXREAL_0:2; ::_thesis: verum
end;
supposeA115: ( i1 < F . (min* domF) & j1 = F . (min* domF) ) ; ::_thesis: min* (PG " {i}) < min* (PG " {j})
 card rF = rng Orde by A42, FUNCT_2:def_3;
then A116: 0 in rng Orde by NAT_1:44;
 (Orde . i1) + 1 in rng Orde by A84, A59, A115;
then A117: Orde9 . ((Orde . i1) + 1) > Orde9 . 0 by A71, A116;
A118: P1 . j1 = Orde9 . 0 by A44, A94, A115;
 Orde9 . ((Orde . i1) + 1) = i by A44, A92, A93, A115;
hence  min* (PG " {i}) < min* (PG " {j}) by A83, A95, A117, A118, TARSKI:def_1; ::_thesis: verum
end;
suppose ( i1 = F . (min* domF) & j1 = F . (min* domF) ) ; ::_thesis: min* (PG " {i}) < min* (PG " {j})
hence  min* (PG " {i}) < min* (PG " {j}) by A83, A95, A93, TARSKI:def_1; ::_thesis: verum
end;
supposeA119: ( i1 > F . (min* domF) & j1 > F . (min* domF) ) ; ::_thesis: min* (PG " {i}) < min* (PG " {j})
A120: ( i1 in rng FD or i1 in {(F . (min* domF))} ) by A85, A80, XBOOLE_0:def_3;
then A121: G " {i1} = PG " {i} by A55, A87, A119, TARSKI:def_1;
A122: P1 . j1 = j1 by A44, A88, A119;
A123: ( j1 in rng FD or j1 in {(F . (min* domF))} ) by A89, A80, XBOOLE_0:def_3;
 P1 . i1 = i1 by A44, A84, A119;
then  min* (G " {i1}) < min* (G " {j1}) by A19, A20, A83, A93, A96, A119, A122, A120, A123, TARSKI:def_1;
hence  min* (PG " {i}) < min* (PG " {j}) by A55, A91, A119, A123, A121, TARSKI:def_1; ::_thesis: verum
end;
supposeA124: ( i1 > F . (min* domF) & j1 = F . (min* domF) ) ; ::_thesis: min* (PG " {i}) < min* (PG " {j})
A125: dom Orde = rF by FUNCT_2:def_1;
 rng (Orde ") = dom Orde by A42, FUNCT_1:33;
then consider x being set  such that
A126: x in dom Orde9 and
A127: Orde9 . x = i1 by A84, A125, FUNCT_1:def_3;
A128: x in card rF by A126;
 card rF is Subset of NAT by Th8;
then reconsider x = x as Element of NAT by A128;
 P1 . i1 = i1 by A44, A84, A124;
then A129: Orde9 . x < Orde9 . 0 by A44, A83, A88, A93, A96, A124, A127;
A130: card rF = rng Orde by A42, FUNCT_2:def_3;
then  0 in rng Orde by NAT_1:44;
then  x <= 0 by A71, A126, A130, A129;
hence  min* (PG " {i}) < min* (PG " {j}) by A129; ::_thesis: verum
end;
supposeA131: ( i1 < F . (min* domF) & j1 > F . (min* domF) ) ; ::_thesis: min* (PG " {i}) < min* (PG " {j})
A132: ( i1 in rng FD or i1 in {(F . (min* domF))} ) by A85, A80, XBOOLE_0:def_3;
then A133: G " {i1} = PG " {i} by A55, A87, A131, TARSKI:def_1;
A134: ( j1 in rng FD or j1 in {(F . (min* domF))} ) by A89, A80, XBOOLE_0:def_3;
 i1 < j1 by A131, XXREAL_0:2;
then  min* (G " {i1}) < min* (G " {j1}) by A19, A20, A131, A132, A134, TARSKI:def_1;
hence  min* (PG " {i}) < min* (PG " {j}) by A55, A91, A131, A134, A133, TARSKI:def_1; ::_thesis: verum
end;
supposeA135: ( i1 > F . (min* domF) & j1 < F . (min* domF) ) ; ::_thesis: min* (PG " {i}) < min* (PG " {j})
then  dom Orde = rF by A88, A59;
then A136: Orde . (F . (min* domF)) in rng Orde by A9, FUNCT_1:def_3;
 (Orde . j1) + 1 <= Orde . (F . (min* domF)) by A88, A59, A135;
then A137: ( (Orde . j1) + 1 < Orde . (F . (min* domF)) or (Orde . j1) + 1 = Orde . (F . (min* domF)) ) by XXREAL_0:1;
 F . (min* domF) in dom Orde by A9, A88, A59, A135;
then A138: Orde9 . (Orde . (F . (min* domF))) = F . (min* domF) by A42, FUNCT_1:34;
A139: P1 . i1 = i1 by A44, A84, A135;
A140: Orde9 . ((Orde . j1) + 1) = P1 . j1 by A44, A88, A135;
 (Orde . j1) + 1 in rng Orde by A88, A59, A135;
then  P1 . j1 <= F . (min* domF) by A71, A137, A136, A138, A140;
hence  min* (PG " {i}) < min* (PG " {j}) by A83, A93, A96, A135, A139, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  min* (PG " {i}) < min* (PG " {j}) ; ::_thesis: verum
end;
 not K is empty ;
then A141: dom F = N by FUNCT_2:def_1;
A142: for x being set st x in dom F holds
F . x = P2 . (G2 . x)
proof
let x be set ; ::_thesis: ( x in dom F implies F . x = P2 . (G2 . x) )
assume A143: x in dom F ; ::_thesis: F . x = P2 . (G2 . x)
now__::_thesis:_F_._x_=_P2_._(G2_._x)
percases ( x in N \ dFD or x in dFD ) by A143, XBOOLE_0:def_5;
supposeA144: x in N \ dFD ; ::_thesis: F . x = P2 . (G2 . x)
then A145: not x in (dom F) \ (F " {(F . (min* domF))}) by A27, XBOOLE_0:def_5;
 x in dom F by A141, A144, XBOOLE_0:def_5;
then  x in F " {(F . (min* domF))} by A145, XBOOLE_0:def_5;
then A146: F . x in {(F . (min* domF))} by FUNCT_1:def_7;
 P2 . (G2 . x) = F . (min* domF) by A22, A40, A144;
hence  F . x = P2 . (G2 . x) by A146, TARSKI:def_1; ::_thesis: verum
end;
supposeA147: x in dFD ; ::_thesis: F . x = P2 . (G2 . x)
then A148: F . x = FD . x by FUNCT_1:47;
A149: FD . x = P . (G . x) by A18, A147, FUNCT_1:12;
A150: dom P = rng FD by FUNCT_2:def_1;
A151: x in dom G by A147, FUNCT_2:def_1;
then A152: G . x in rng FD by A19, FUNCT_1:def_3;
 G . x = G2 . x by A22, A151, FUNCT_1:47;
hence  F . x = P2 . (G2 . x) by A39, A148, A149, A152, A150, FUNCT_1:47; ::_thesis: verum
end;
end;
end;
hence  F . x = P2 . (G2 . x) ; ::_thesis: verum
end;
A153: dom G2 = N by FUNCT_2:def_1;
 for x being set holds
( x in dom F iff ( x in dom G2 & G2 . x in dom P2 ) )
proof
let x be set ; ::_thesis: ( x in dom F iff ( x in dom G2 & G2 . x in dom P2 ) )
thus  ( x in dom F implies ( x in dom G2 & G2 . x in dom P2 ) ) ::_thesis: ( x in dom G2 & G2 . x in dom P2 implies x in dom F )
proof
assume A154: x in dom F ; ::_thesis: ( x in dom G2 & G2 . x in dom P2 )
then  dom P2 = rng F by A153, FUNCT_2:def_1;
hence  ( x in dom G2 & G2 . x in dom P2 ) by A38, A153, A154, FUNCT_1:def_3; ::_thesis: verum
end;
thus  ( x in dom G2 & G2 . x in dom P2 implies x in dom F ) by A141; ::_thesis: verum
end;
then  F = P2 * G2 by A142, FUNCT_1:10;
then A155: F = P21 * PG by A47, RELAT_1:36;
 rng P1 = rF by FUNCT_2:def_3;
then  rng (P1 * G2) = rF by A38, A45, FUNCT_2:14;
hence  ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) ) by A155, A58; ::_thesis: verum
end;
end;
end;
hence  ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) ) ; ::_thesis: verum
end;
end;
end;
hence  ex P being Permutation of (rng F) ex G being Function of N,K st
( F = P * G & rng F = rng G & ( for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) ) ) ; ::_thesis: verum
end;
A156: for F being Function st rng F is finite holds
S2[F] from STIRL2_1:sch_7(A1, A4);
let F be Function of Ne,Ke; ::_thesis: ( rng F is finite implies ex I being Function of Ne,Ke ex P being Permutation of (rng F) st
( F = P * I & rng F = rng I & I is "increasing ) )
assume  rng F is finite ; ::_thesis: ex I being Function of Ne,Ke ex P being Permutation of (rng F) st
( F = P * I & rng F = rng I & I is "increasing )
then consider P being Permutation of (rng F), G being Function of Ne,Ke such that
A157: F = P * G and
A158: rng F = rng G and
A159: for i, j being Nat st i in rng G & j in rng G & i < j holds
min* (G " {i}) < min* (G " {j}) by A156;
 G is "increasing by A159, Def3;
hence  ex I being Function of Ne,Ke ex P being Permutation of (rng F) st
( F = P * I & rng F = rng I & I is "increasing ) by A157, A158; ::_thesis: verum
end;

theorem Th64: :: STIRL2_1:64
for Ne, Ke, Me being Subset of NAT
for F being Function of Ne,Ke st rng F is finite holds
for I1, I2 being Function of Ne,Me
for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
proof
let Ne, Ke, Me be Subset of NAT; ::_thesis: for F being Function of Ne,Ke st rng F is finite holds
for I1, I2 being Function of Ne,Me
for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
defpred S1[ set , Function] means $1 = $2 . (min* (dom $2));
defpred S2[ set ] means  for Ne, Ke, Me being Subset of NAT
for F being Function of Ne,Ke st F = $1 holds
for I1, I2 being Function of Ne,Me
for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 );
A1: S2[ {} ]
proof
let Ne, Ke, Me be Subset of NAT; ::_thesis: for F being Function of Ne,Ke st F = {} holds
for I1, I2 being Function of Ne,Me
for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
let F be Function of Ne,Ke; ::_thesis: ( F = {} implies for I1, I2 being Function of Ne,Me
for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 ) )
assume A2: F = {} ; ::_thesis: for I1, I2 being Function of Ne,Me
for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
let I1, I2 be Function of Ne,Me; ::_thesis: for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
let P1, P2 be Function; ::_thesis: ( P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing implies ( P1 = P2 & I1 = I2 ) )
assume that
 P1 is one-to-one and
 P2 is one-to-one and
A3: rng I1 = rng I2 and
A4: rng I1 = dom P1 and
A5: dom P1 = dom P2 and
A6: F = P1 * I1 and
 F = P2 * I2 and
 I1 is "increasing and
 I2 is "increasing ; ::_thesis: ( P1 = P2 & I1 = I2 )
 dom I1 = {} by A2, A4, A6, RELAT_1:27, RELAT_1:38;
then A7: I1 = {} ;
 rng P1 = {} by A2, A4, A6, RELAT_1:28, RELAT_1:38;
then  P1 = {} ;
hence  ( P1 = P2 & I1 = I2 ) by A3, A5, A7; ::_thesis: verum
end;
A8: for F being Function st ( for x being set st x in rng F & S1[x,F] holds
S2[F | ((dom F) \ (F " {x}))] ) holds
S2[F]
proof
let F9 be Function; ::_thesis: ( ( for x being set st x in rng F9 & S1[x,F9] holds
S2[F9 | ((dom F9) \ (F9 " {x}))] ) implies S2[F9] )
assume A9: for x being set st x in rng F9 & S1[x,F9] holds
S2[F9 | ((dom F9) \ (F9 " {x}))] ; ::_thesis: S2[F9]
now__::_thesis:_S2[F9]
percases ( F9 = {} or F9 <> {} ) ;
suppose F9 = {} ; ::_thesis: S2[F9]
hence  S2[F9] by A1; ::_thesis: verum
end;
supposeA10: F9 <> {} ; ::_thesis: for Ne, Ke, Me being Subset of NAT
for F being Function of Ne,Ke st F = F9 holds
for I1, I2 being Function of Ne,Me
for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
let Ne, Ke, Me be Subset of NAT; ::_thesis: for F being Function of Ne,Ke st F = F9 holds
for I1, I2 being Function of Ne,Me
for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
let F be Function of Ne,Ke; ::_thesis: ( F = F9 implies for I1, I2 being Function of Ne,Me
for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 ) )
assume A11: F = F9 ; ::_thesis: for I1, I2 being Function of Ne,Me
for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
 not Ke is empty by A10, A11;
then reconsider domF = dom F as non empty Subset of NAT by A10, A11, FUNCT_2:def_1;
set m = min* (dom F);
let I1, I2 be Function of Ne,Me; ::_thesis: for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
let P1, P2 be Function; ::_thesis: ( P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing implies ( P1 = P2 & I1 = I2 ) )
assume that
A12: P1 is one-to-one and
A13: P2 is one-to-one and
A14: rng I1 = rng I2 and
A15: rng I1 = dom P1 and
A16: dom P1 = dom P2 and
A17: F = P1 * I1 and
A18: F = P2 * I2 and
A19: I1 is "increasing and
A20: I2 is "increasing ; ::_thesis: ( P1 = P2 & I1 = I2 )
 dom I1 = dom F by A15, A17, RELAT_1:27;
then A21: min* (rng I1) = I1 . (min* (dom F)) by A19, Th62;
reconsider I = (rng I1) \ {(I1 . (min* (dom F)))} as Subset of NAT by XBOOLE_1:1;
reconsider D = (dom F) \ (F " {(F . (min* (dom F)))}) as Subset of NAT by XBOOLE_1:1;
A22: for I being Function of Ne,Me
for P being Function st P is one-to-one & rng I = dom P & F = P * I & I is "increasing holds
( dom (I | D) = D & rng (I | D) = (rng I) \ {(I . (min* (dom F)))} & dom (P | ((rng I) \ {(I . (min* (dom F)))})) = (rng I) \ {(I . (min* (dom F)))} & F | D = (P | ((rng I) \ {(I . (min* (dom F)))})) * (I | D) & P | ((rng I) \ {(I . (min* (dom F)))}) is one-to-one & ( for M being Subset of NAT st M = (rng I) \ {(I . (min* (dom F)))} holds
for I1 being Function of D,M st I1 = I | D holds
I1 is "increasing ) )
proof
A23: F . (min* (dom F)) in {(F . (min* (dom F)))} by TARSKI:def_1;
let I be Function of Ne,Me; ::_thesis: for P being Function st P is one-to-one & rng I = dom P & F = P * I & I is "increasing holds
( dom (I | D) = D & rng (I | D) = (rng I) \ {(I . (min* (dom F)))} & dom (P | ((rng I) \ {(I . (min* (dom F)))})) = (rng I) \ {(I . (min* (dom F)))} & F | D = (P | ((rng I) \ {(I . (min* (dom F)))})) * (I | D) & P | ((rng I) \ {(I . (min* (dom F)))}) is one-to-one & ( for M being Subset of NAT st M = (rng I) \ {(I . (min* (dom F)))} holds
for I1 being Function of D,M st I1 = I | D holds
I1 is "increasing ) )
let P be Function; ::_thesis: ( P is one-to-one & rng I = dom P & F = P * I & I is "increasing implies ( dom (I | D) = D & rng (I | D) = (rng I) \ {(I . (min* (dom F)))} & dom (P | ((rng I) \ {(I . (min* (dom F)))})) = (rng I) \ {(I . (min* (dom F)))} & F | D = (P | ((rng I) \ {(I . (min* (dom F)))})) * (I | D) & P | ((rng I) \ {(I . (min* (dom F)))}) is one-to-one & ( for M being Subset of NAT st M = (rng I) \ {(I . (min* (dom F)))} holds
for I1 being Function of D,M st I1 = I | D holds
I1 is "increasing ) ) )
assume that
A24: P is one-to-one and
A25: rng I = dom P and
A26: F = P * I and
A27: I is "increasing ; ::_thesis: ( dom (I | D) = D & rng (I | D) = (rng I) \ {(I . (min* (dom F)))} & dom (P | ((rng I) \ {(I . (min* (dom F)))})) = (rng I) \ {(I . (min* (dom F)))} & F | D = (P | ((rng I) \ {(I . (min* (dom F)))})) * (I | D) & P | ((rng I) \ {(I . (min* (dom F)))}) is one-to-one & ( for M being Subset of NAT st M = (rng I) \ {(I . (min* (dom F)))} holds
for I1 being Function of D,M st I1 = I | D holds
I1 is "increasing ) )
A28: (dom P) /\ ((rng I) \ {(I . (min* (dom F)))}) = (rng I) \ {(I . (min* (dom F)))} by A25, XBOOLE_1:28, XBOOLE_1:36;
 min* (dom F) in domF by NAT_1:def_1;
then  F . (min* (dom F)) in rng F by FUNCT_1:def_3;
then consider x being set  such that
 x in dom P and
 x in rng I and
 P " {(F . (min* (dom F)))} = {x} and
A29: I " {x} = F " {(F . (min* (dom F)))} by A24, A26, Th61;
A30: dom (P | ((rng I) \ {(I . (min* (dom F)))})) = (dom P) /\ ((rng I) \ {(I . (min* (dom F)))}) by RELAT_1:61;
A31: dom F = dom I by A25, A26, RELAT_1:27;
then A32: (dom I) /\ D = D by XBOOLE_1:28, XBOOLE_1:36;
 min* (dom F) in domF by NAT_1:def_1;
then  min* (dom F) in I " {x} by A29, A23, FUNCT_1:def_7;
then  I . (min* (dom F)) in {x} by FUNCT_1:def_7;
then A33: I . (min* (dom F)) = x by TARSKI:def_1;
A34: for M being Subset of NAT st M = (rng I) \ {(I . (min* (dom F)))} holds
for I1 being Function of D,M st I1 = I | D holds
I1 is "increasing
proof
let M be Subset of NAT; ::_thesis: ( M = (rng I) \ {(I . (min* (dom F)))} implies for I1 being Function of D,M st I1 = I | D holds
I1 is "increasing )
assume  M = (rng I) \ {(I . (min* (dom F)))} ; ::_thesis: for I1 being Function of D,M st I1 = I | D holds
I1 is "increasing
let I1 be Function of D,M; ::_thesis: ( I1 = I | D implies I1 is "increasing )
assume A35: I1 = I | D ; ::_thesis: I1 is "increasing
A36: rng I1 = (rng I) \ {(I . (min* (dom F)))} by A29, A33, A31, A35, Th54;
let l be Nat; :: according to STIRL2_1:def_3 ::_thesis: for m being Nat st l in rng I1 & m in rng I1 & l < m holds
min* (I1 " {l}) < min* (I1 " {m})
let n be Nat; ::_thesis: ( l in rng I1 & n in rng I1 & l < n implies min* (I1 " {l}) < min* (I1 " {n}) )
assume that
A37: l in rng I1 and
A38: n in rng I1 and
A39: l < n ; ::_thesis: min* (I1 " {l}) < min* (I1 " {n})
A40: n in rng I by A38, A36, ZFMISC_1:56;
 n <> I . (min* (dom F)) by A38, A36, ZFMISC_1:56;
then A41: I1 " {n} = I " {n} by A29, A33, A31, A35, Th54;
 l <> I . (min* (dom F)) by A37, A36, ZFMISC_1:56;
then A42: I1 " {l} = I " {l} by A29, A33, A31, A35, Th54;
 l in rng I by A37, A36, ZFMISC_1:56;
hence  min* (I1 " {l}) < min* (I1 " {n}) by A27, A39, A40, A42, A41, Def3; ::_thesis: verum
end;
set rI = (rng I) \ {(I . (min* (dom F)))};
A43: dom (I | D) = (dom I) /\ D by RELAT_1:61;
A44: rng (I | D) = (rng I) \ {(I . (min* (dom F)))} by A29, A33, A31, Th54;
A45: for x being set st x in dom (F | D) holds
(F | D) . x = (P | ((rng I) \ {(I . (min* (dom F)))})) . ((I | D) . x)
proof
let x be set ; ::_thesis: ( x in dom (F | D) implies (F | D) . x = (P | ((rng I) \ {(I . (min* (dom F)))})) . ((I | D) . x) )
assume A46: x in dom (F | D) ; ::_thesis: (F | D) . x = (P | ((rng I) \ {(I . (min* (dom F)))})) . ((I | D) . x)
A47: x in (dom F) /\ D by A46, RELAT_1:61;
then A48: x in dom F by XBOOLE_0:def_4;
 (I | D) . x in dom (P | ((rng I) \ {(I . (min* (dom F)))})) by A31, A43, A30, A28, A44, A47, FUNCT_1:def_3;
then A49: (P | ((rng I) \ {(I . (min* (dom F)))})) . ((I | D) . x) = P . ((I | D) . x) by FUNCT_1:47;
A50: (F | D) . x = F . x by A46, FUNCT_1:47;
 I . x = (I | D) . x by A31, A43, A47, FUNCT_1:47;
hence  (F | D) . x = (P | ((rng I) \ {(I . (min* (dom F)))})) . ((I | D) . x) by A26, A48, A49, A50, FUNCT_1:12; ::_thesis: verum
end;
 (dom F) /\ D = D by XBOOLE_1:28, XBOOLE_1:36;
then  dom (F | D) = D by RELAT_1:61;
then  for x being set holds
( x in dom (F | D) iff ( x in dom (I | D) & (I | D) . x in dom (P | ((rng I) \ {(I . (min* (dom F)))})) ) ) by A43, A32, A30, A28, A44, FUNCT_1:def_3;
hence  ( dom (I | D) = D & rng (I | D) = (rng I) \ {(I . (min* (dom F)))} & dom (P | ((rng I) \ {(I . (min* (dom F)))})) = (rng I) \ {(I . (min* (dom F)))} & F | D = (P | ((rng I) \ {(I . (min* (dom F)))})) * (I | D) & P | ((rng I) \ {(I . (min* (dom F)))}) is one-to-one & ( for M being Subset of NAT st M = (rng I) \ {(I . (min* (dom F)))} holds
for I1 being Function of D,M st I1 = I | D holds
I1 is "increasing ) ) by A24, A29, A33, A31, A32, A28, A45, A34, Th54, FUNCT_1:10, FUNCT_1:52, RELAT_1:61; ::_thesis: verum
end;
then A51: P1 | I is one-to-one by A12, A15, A17, A19;
 dom I2 = dom F by A14, A15, A16, A18, RELAT_1:27;
then A52: min* (rng I2) = I2 . (min* (dom F)) by A20, Th62;
then A53: P2 | I is one-to-one by A13, A14, A15, A16, A18, A20, A22, A21;
A54: dom (I2 | D) = D by A13, A14, A15, A16, A18, A20, A22;
A55: dom (I1 | D) = D by A12, A15, A17, A19, A22;
A56: rng (I1 | D) = I by A12, A15, A17, A19, A22;
 rng (I2 | D) = I by A13, A14, A15, A16, A18, A20, A22, A21, A52;
then reconsider I1D = I1 | D, I2D = I2 | D as Function of D,I by A55, A54, A56, FUNCT_2:1;
A57: I2D is "increasing by A13, A14, A15, A16, A18, A20, A22, A21, A52;
A58: rng I1D = I by A12, A15, A17, A19, A22;
then A59: rng I1D = dom (P1 | I) by A12, A15, A17, A19, A22;
reconsider rFD = rng (F | D) as Subset of NAT by XBOOLE_1:1;
 (dom F) /\ D = D by XBOOLE_1:28, XBOOLE_1:36;
then  dom (F | D) = D by RELAT_1:61;
then reconsider FD = F | D as Function of D,rFD by FUNCT_2:1;
A60: FD = (P1 | I) * I1D by A12, A15, A17, A19, A22;
A61: FD = (P2 | I) * I2D by A13, A14, A15, A16, A18, A20, A22, A21, A52;
 min* (dom F) in domF by NAT_1:def_1;
then A62: F . (min* (dom F)) in rng F by FUNCT_1:def_3;
 dom (P1 | I) = I by A12, A15, A17, A19, A22;
then A63: dom (P1 | I) = dom (P2 | I) by A13, A14, A15, A16, A18, A20, A22, A21, A52;
A64: I1D is "increasing by A12, A15, A17, A19, A22;
A65: rng I1D = rng I2D by A13, A14, A15, A16, A18, A20, A22, A21, A52, A58;
 for x being set st x in dom P1 holds
P1 . x = P2 . x
proof
A66: min* (dom F) in domF by NAT_1:def_1;
 dom I1 = dom F by A15, A17, RELAT_1:27;
then  I1 . (min* (dom F)) in rng I1 by A66, FUNCT_1:def_3;
then A67: dom P1 = I \/ {(I1 . (min* (dom F)))} by A15, ZFMISC_1:116;
let x be set ; ::_thesis: ( x in dom P1 implies P1 . x = P2 . x )
assume A68: x in dom P1 ; ::_thesis: P1 . x = P2 . x
now__::_thesis:_P1_._x_=_P2_._x
percases ( x in I or x in {(I1 . (min* (dom F)))} ) by A68, A67, XBOOLE_0:def_3;
supposeA69: x in I ; ::_thesis: P1 . x = P2 . x
 (dom P1) /\ I = I by A15, XBOOLE_1:28, XBOOLE_1:36;
then  x in dom (P1 | I) by A69, RELAT_1:61;
then A70: (P1 | I) . x = P1 . x by FUNCT_1:47;
 (dom P2) /\ I = I by A15, A16, XBOOLE_1:28, XBOOLE_1:36;
then  x in dom (P2 | I) by A69, RELAT_1:61;
then  (P2 | I) . x = P2 . x by FUNCT_1:47;
hence  P1 . x = P2 . x by A9, A11, A62, A51, A53, A65, A59, A63, A60, A61, A64, A57, A70; ::_thesis: verum
end;
supposeA71: x in {(I1 . (min* (dom F)))} ; ::_thesis: P1 . x = P2 . x
A72: min* (dom F) in domF by NAT_1:def_1;
A73: x = I1 . (min* (dom F)) by A71, TARSKI:def_1;
then  F . (min* (dom F)) = P1 . x by A17, A72, FUNCT_1:12;
hence  P1 . x = P2 . x by A14, A18, A21, A52, A73, A72, FUNCT_1:12; ::_thesis: verum
end;
end;
end;
hence  P1 . x = P2 . x ; ::_thesis: verum
end;
then A74: P1 = P2 by A16, FUNCT_1:def_11;
 I2 is Function of (dom I2),(rng I2) by FUNCT_2:1;
then A75: I2 = (id (rng I2)) * I2 by FUNCT_2:17;
 I1 is Function of (dom I1),(rng I1) by FUNCT_2:1;
then A76: I1 = (id (rng I1)) * I1 by FUNCT_2:17;
 (P1 ") * P1 = id (dom P1) by A12, FUNCT_1:39;
then A77: I1 = (P1 ") * (P1 * I1) by A15, A76, RELAT_1:36;
 (P2 ") * P2 = id (dom P2) by A13, FUNCT_1:39;
hence  ( P1 = P2 & I1 = I2 ) by A14, A15, A17, A18, A74, A75, A77, RELAT_1:36; ::_thesis: verum
end;
end;
end;
hence  S2[F9] ; ::_thesis: verum
end;
 for F being Function st rng F is finite holds
S2[F] from STIRL2_1:sch_7(A1, A8);
hence  for F being Function of Ne,Ke st rng F is finite holds
for I1, I2 being Function of Ne,Me
for P1, P2 being Function st P1 is one-to-one & P2 is one-to-one & rng I1 = rng I2 & rng I1 = dom P1 & dom P1 = dom P2 & F = P1 * I1 & F = P2 * I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 ) ; ::_thesis: verum
end;

theorem :: STIRL2_1:65
for Ne, Ke being Subset of NAT
for F being Function of Ne,Ke st rng F is finite holds
for I1, I2 being Function of Ne,Ke
for P1, P2 being Permutation of (rng F) st F = P1 * I1 & F = P2 * I2 & rng F = rng I1 & rng F = rng I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
proof
let Ne, Ke be Subset of NAT; ::_thesis: for F being Function of Ne,Ke st rng F is finite holds
for I1, I2 being Function of Ne,Ke
for P1, P2 being Permutation of (rng F) st F = P1 * I1 & F = P2 * I2 & rng F = rng I1 & rng F = rng I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
let F be Function of Ne,Ke; ::_thesis: ( rng F is finite implies for I1, I2 being Function of Ne,Ke
for P1, P2 being Permutation of (rng F) st F = P1 * I1 & F = P2 * I2 & rng F = rng I1 & rng F = rng I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 ) )
assume A1: rng F is finite ; ::_thesis: for I1, I2 being Function of Ne,Ke
for P1, P2 being Permutation of (rng F) st F = P1 * I1 & F = P2 * I2 & rng F = rng I1 & rng F = rng I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
let I1, I2 be Function of Ne,Ke; ::_thesis: for P1, P2 being Permutation of (rng F) st F = P1 * I1 & F = P2 * I2 & rng F = rng I1 & rng F = rng I2 & I1 is "increasing & I2 is "increasing holds
( P1 = P2 & I1 = I2 )
let P1, P2 be Permutation of (rng F); ::_thesis: ( F = P1 * I1 & F = P2 * I2 & rng F = rng I1 & rng F = rng I2 & I1 is "increasing & I2 is "increasing implies ( P1 = P2 & I1 = I2 ) )
assume that
A2: F = P1 * I1 and
A3: F = P2 * I2 and
A4: rng F = rng I1 and
A5: rng F = rng I2 and
A6: I1 is "increasing and
A7: I2 is "increasing ; ::_thesis: ( P1 = P2 & I1 = I2 )
A8: ( rng F = {} implies rng F = {} ) ;
then A9: dom P2 = rng F by FUNCT_2:def_1;
 dom P1 = rng F by A8, FUNCT_2:def_1;
hence  ( P1 = P2 & I1 = I2 ) by A1, A2, A3, A4, A5, A6, A7, A9, Th64; ::_thesis: verum
end;

theorem :: STIRL2_1:66
for D being non empty set
for F being XFinSequence of D st ( for i being Nat st i in dom F holds
F . i is finite ) & ( for i, j being Nat st i in dom F & j in dom F & i <> j holds
F . i misses F . j ) holds
ex CardF being XFinSequence of NAT st
( dom CardF = dom F & ( for i being Nat st i in dom CardF holds
CardF . i = card (F . i) ) & card (union (rng F)) = Sum CardF ) by Lm2;