:: FUNCT_7 semantic presentation

begin


theorem :: FUNCT_7:1
for f being Function
for X being set st rng f c= X holds
(id X) * f = f
proof
let f be Function; ::_thesis: for X being set st rng f c= X holds
(id X) * f = f
let X be set ; ::_thesis: ( rng f c= X implies (id X) * f = f )
assume  rng f c= X ; ::_thesis: (id X) * f = f
then reconsider g = f as Function of (dom f),X by FUNCT_2:2;
 (id X) * g = g by FUNCT_2:17;
hence  (id X) * f = f ; ::_thesis: verum
end;

theorem :: FUNCT_7:2
for X being set
for Y being non empty set
for f being Function of X,Y st f is one-to-one holds
for B being Subset of X
for C being Subset of Y st C c= f .: B holds
f " C c= B
proof
let X be set ; ::_thesis: for Y being non empty set
for f being Function of X,Y st f is one-to-one holds
for B being Subset of X
for C being Subset of Y st C c= f .: B holds
f " C c= B
let Y be non empty set ; ::_thesis: for f being Function of X,Y st f is one-to-one holds
for B being Subset of X
for C being Subset of Y st C c= f .: B holds
f " C c= B
let f be Function of X,Y; ::_thesis: ( f is one-to-one implies for B being Subset of X
for C being Subset of Y st C c= f .: B holds
f " C c= B )
assume A1: f is one-to-one ; ::_thesis: for B being Subset of X
for C being Subset of Y st C c= f .: B holds
f " C c= B
let B be Subset of X; ::_thesis: for C being Subset of Y st C c= f .: B holds
f " C c= B
let C be Subset of Y; ::_thesis: ( C c= f .: B implies f " C c= B )
assume  C c= f .: B ; ::_thesis: f " C c= B
then A2: f " C c= f " (f .: B) by RELAT_1:143;
 f " (f .: B) c= B by A1, FUNCT_1:82;
hence  f " C c= B by A2, XBOOLE_1:1; ::_thesis: verum
end;

theorem Th3: :: FUNCT_7:3
for X, Y being non empty set
for f being Function of X,Y st f is one-to-one holds
for x being Element of X
for A being Subset of X st f . x in f .: A holds
x in A
proof
let X, Y be non empty set ; ::_thesis: for f being Function of X,Y st f is one-to-one holds
for x being Element of X
for A being Subset of X st f . x in f .: A holds
x in A
let f be Function of X,Y; ::_thesis: ( f is one-to-one implies for x being Element of X
for A being Subset of X st f . x in f .: A holds
x in A )
assume A1: f is one-to-one ; ::_thesis: for x being Element of X
for A being Subset of X st f . x in f .: A holds
x in A
let x be Element of X; ::_thesis: for A being Subset of X st f . x in f .: A holds
x in A
let A be Subset of X; ::_thesis: ( f . x in f .: A implies x in A )
assume  f . x in f .: A ; ::_thesis: x in A
then  ex y being Element of X st
( y in A & f . y = f . x ) by FUNCT_2:65;
hence  x in A by A1, FUNCT_2:19; ::_thesis: verum
end;

theorem Th4: :: FUNCT_7:4
for X, Y being non empty set
for f being Function of X,Y st f is one-to-one holds
for x being Element of X
for A being Subset of X
for B being Subset of Y st f . x in (f .: A) \ B holds
x in A \ (f " B)
proof
let X, Y be non empty set ; ::_thesis: for f being Function of X,Y st f is one-to-one holds
for x being Element of X
for A being Subset of X
for B being Subset of Y st f . x in (f .: A) \ B holds
x in A \ (f " B)
let f be Function of X,Y; ::_thesis: ( f is one-to-one implies for x being Element of X
for A being Subset of X
for B being Subset of Y st f . x in (f .: A) \ B holds
x in A \ (f " B) )
assume A1: f is one-to-one ; ::_thesis: for x being Element of X
for A being Subset of X
for B being Subset of Y st f . x in (f .: A) \ B holds
x in A \ (f " B)
let x be Element of X; ::_thesis: for A being Subset of X
for B being Subset of Y st f . x in (f .: A) \ B holds
x in A \ (f " B)
let A be Subset of X; ::_thesis: for B being Subset of Y st f . x in (f .: A) \ B holds
x in A \ (f " B)
let B be Subset of Y; ::_thesis: ( f . x in (f .: A) \ B implies x in A \ (f " B) )
assume A2: f . x in (f .: A) \ B ; ::_thesis: x in A \ (f " B)
A3: now__::_thesis:_not_x_in_f_"_B
assume  x in f " B ; ::_thesis: contradiction
then  f . x in B by FUNCT_1:def_7;
hence  contradiction by A2, XBOOLE_0:def_5; ::_thesis: verum
end;
 f . x in f .: A by A2, XBOOLE_0:def_5;
then  x in A by A1, Th3;
hence  x in A \ (f " B) by A3, XBOOLE_0:def_5; ::_thesis: verum
end;

theorem :: FUNCT_7:5
for X, Y being non empty set
for f being Function of X,Y st f is one-to-one holds
for y being Element of Y
for A being Subset of X
for B being Subset of Y st y in (f .: A) \ B holds
(f ") . y in A \ (f " B)
proof
let X, Y be non empty set ; ::_thesis: for f being Function of X,Y st f is one-to-one holds
for y being Element of Y
for A being Subset of X
for B being Subset of Y st y in (f .: A) \ B holds
(f ") . y in A \ (f " B)
let f be Function of X,Y; ::_thesis: ( f is one-to-one implies for y being Element of Y
for A being Subset of X
for B being Subset of Y st y in (f .: A) \ B holds
(f ") . y in A \ (f " B) )
assume A1: f is one-to-one ; ::_thesis: for y being Element of Y
for A being Subset of X
for B being Subset of Y st y in (f .: A) \ B holds
(f ") . y in A \ (f " B)
let y be Element of Y; ::_thesis: for A being Subset of X
for B being Subset of Y st y in (f .: A) \ B holds
(f ") . y in A \ (f " B)
let A be Subset of X; ::_thesis: for B being Subset of Y st y in (f .: A) \ B holds
(f ") . y in A \ (f " B)
let B be Subset of Y; ::_thesis: ( y in (f .: A) \ B implies (f ") . y in A \ (f " B) )
assume A2: y in (f .: A) \ B ; ::_thesis: (f ") . y in A \ (f " B)
then A3: y in f .: A by XBOOLE_0:def_5;
A4: f .: A c= rng f by RELAT_1:111;
then  (f ") . y in dom f by A1, A3, FUNCT_1:32;
then reconsider x = (f ") . y as Element of X ;
 y = f . x by A1, A3, A4, FUNCT_1:35;
hence  (f ") . y in A \ (f " B) by A1, A2, Th4; ::_thesis: verum
end;

theorem Th6: :: FUNCT_7:6
for f being Function
for a being set st a in dom f holds
f | {a} = a .--> (f . a)
proof
let f be Function; ::_thesis: for a being set st a in dom f holds
f | {a} = a .--> (f . a)
let a be set ; ::_thesis: ( a in dom f implies f | {a} = a .--> (f . a) )
assume  a in dom f ; ::_thesis: f | {a} = a .--> (f . a)
hence f | {a} = {[a,(f . a)]} by GRFUNC_1:28
.= a .--> (f . a) by ZFMISC_1:29 ;
::_thesis: verum
end;

registration
let x, y be set ;
clusterx .--> y ->  non empty ;
coherence
 not x .--> y is empty ;
end;

registration
let x, y, a, b be set ;
cluster(x,y) --> (a,b) ->  non empty ;
coherence
 not (x,y) --> (a,b) is empty ;
end;

theorem Th7: :: FUNCT_7:7
for I being set
for M being ManySortedSet of I
for i being set st i in I holds
i .--> (M . i) = M | {i}
proof
let I be set ; ::_thesis: for M being ManySortedSet of I
for i being set st i in I holds
i .--> (M . i) = M | {i}
let M be ManySortedSet of I; ::_thesis: for i being set st i in I holds
i .--> (M . i) = M | {i}
let i be set ; ::_thesis: ( i in I implies i .--> (M . i) = M | {i} )
assume  i in I ; ::_thesis: i .--> (M . i) = M | {i}
then  i in dom M by PARTFUN1:def_2;
hence  i .--> (M . i) = M | {i} by Th6; ::_thesis: verum
end;

theorem :: FUNCT_7:8
for I, J being set
for M being ManySortedSet of [:I,J:]
for i, j being set st i in I & j in J holds
(i,j) :-> (M . (i,j)) = M | [:{i},{j}:]
proof
let I, J be set ; ::_thesis: for M being ManySortedSet of [:I,J:]
for i, j being set st i in I & j in J holds
(i,j) :-> (M . (i,j)) = M | [:{i},{j}:]
let M be ManySortedSet of [:I,J:]; ::_thesis: for i, j being set st i in I & j in J holds
(i,j) :-> (M . (i,j)) = M | [:{i},{j}:]
let i, j be set ; ::_thesis: ( i in I & j in J implies (i,j) :-> (M . (i,j)) = M | [:{i},{j}:] )
assume  ( i in I & j in J ) ; ::_thesis: (i,j) :-> (M . (i,j)) = M | [:{i},{j}:]
then A1: [i,j] in [:I,J:] by ZFMISC_1:87;
thus (i,j) :-> (M . (i,j)) = [i,j] .--> (M . [i,j])
.= M | {[i,j]} by A1, Th7
.= M | [:{i},{j}:] by ZFMISC_1:29 ; ::_thesis: verum
end;

theorem Th9: :: FUNCT_7:9
for f, g, h being Function st rng h c= dom f holds
f * (g +* h) = (f * g) +* (f * h)
proof
let f, g, h be Function; ::_thesis: ( rng h c= dom f implies f * (g +* h) = (f * g) +* (f * h) )
assume A1: rng h c= dom f ; ::_thesis: f * (g +* h) = (f * g) +* (f * h)
A2: dom h c= dom (g +* h) by FUNCT_4:10;
A3: dom g c= dom (g +* h) by FUNCT_4:10;
A4: dom (f * (g +* h)) = (dom (f * g)) \/ (dom (f * h))
proof
thus  dom (f * (g +* h)) c= (dom (f * g)) \/ (dom (f * h)) :: according to XBOOLE_0:def_10 ::_thesis: (dom (f * g)) \/ (dom (f * h)) c= dom (f * (g +* h))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (f * (g +* h)) or x in (dom (f * g)) \/ (dom (f * h)) )
assume A5: x in dom (f * (g +* h)) ; ::_thesis: x in (dom (f * g)) \/ (dom (f * h))
then A6: (g +* h) . x in dom f by FUNCT_1:11;
 x in dom (g +* h) by A5, FUNCT_1:11;
then A7: x in (dom g) \/ (dom h) by FUNCT_4:def_1;
percases ( x in dom h or not x in dom h ) ;
supposeA8: x in dom h ; ::_thesis: x in (dom (f * g)) \/ (dom (f * h))
then  h . x in dom f by A6, FUNCT_4:13;
then  x in dom (f * h) by A8, FUNCT_1:11;
hence  x in (dom (f * g)) \/ (dom (f * h)) by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose not x in dom h ; ::_thesis: x in (dom (f * g)) \/ (dom (f * h))
then  ( g . x in dom f & x in dom g ) by A7, A6, FUNCT_4:11, XBOOLE_0:def_3;
then  x in dom (f * g) by FUNCT_1:11;
hence  x in (dom (f * g)) \/ (dom (f * h)) by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (dom (f * g)) \/ (dom (f * h)) or x in dom (f * (g +* h)) )
assume A9: x in (dom (f * g)) \/ (dom (f * h)) ; ::_thesis: x in dom (f * (g +* h))
percases ( x in dom (f * h) or not x in dom (f * h) ) ;
supposeA10: x in dom (f * h) ; ::_thesis: x in dom (f * (g +* h))
then A11: h . x in dom f by FUNCT_1:11;
A12: x in dom h by A10, FUNCT_1:11;
then  (g +* h) . x = h . x by FUNCT_4:13;
hence  x in dom (f * (g +* h)) by A2, A12, A11, FUNCT_1:11; ::_thesis: verum
end;
supposeA13: not x in dom (f * h) ; ::_thesis: x in dom (f * (g +* h))
 ( x in dom h implies h . x in dom f )
proof
assume  x in dom h ; ::_thesis: h . x in dom f
then  h . x in rng h by FUNCT_1:3;
hence  h . x in dom f by A1; ::_thesis: verum
end;
then  not x in dom h by A13, FUNCT_1:11;
then A14: (g +* h) . x = g . x by FUNCT_4:11;
 x in dom (f * g) by A9, A13, XBOOLE_0:def_3;
then  ( g . x in dom f & x in dom g ) by FUNCT_1:11;
hence  x in dom (f * (g +* h)) by A3, A14, FUNCT_1:11; ::_thesis: verum
end;
end;
end;
now__::_thesis:_for_x_being_set_st_x_in_(dom_(f_*_g))_\/_(dom_(f_*_h))_holds_
(_(_x_in_dom_(f_*_h)_implies_(f_*_(g_+*_h))_._x_=_(f_*_h)_._x_)_&_(_not_x_in_dom_(f_*_h)_implies_(f_*_(g_+*_h))_._x_=_(f_*_g)_._x_)_)
let x be set ; ::_thesis: ( x in (dom (f * g)) \/ (dom (f * h)) implies ( ( x in dom (f * h) implies (f * (g +* h)) . x = (f * h) . x ) & ( not x in dom (f * h) implies (f * (g +* h)) . x = (f * g) . x ) ) )
assume A15: x in (dom (f * g)) \/ (dom (f * h)) ; ::_thesis: ( ( x in dom (f * h) implies (f * (g +* h)) . x = (f * h) . x ) & ( not x in dom (f * h) implies (f * (g +* h)) . x = (f * g) . x ) )
thus  ( x in dom (f * h) implies (f * (g +* h)) . x = (f * h) . x ) ::_thesis: ( not x in dom (f * h) implies (f * (g +* h)) . x = (f * g) . x )
proof
assume  x in dom (f * h) ; ::_thesis: (f * (g +* h)) . x = (f * h) . x
then A16: x in dom h by FUNCT_1:11;
hence (f * (g +* h)) . x = f . ((g +* h) . x) by A2, FUNCT_1:13
.= f . (h . x) by A16, FUNCT_4:13
.= (f * h) . x by A16, FUNCT_1:13 ;
::_thesis: verum
end;
assume A17: not x in dom (f * h) ; ::_thesis: (f * (g +* h)) . x = (f * g) . x
 ( x in dom h implies h . x in dom f )
proof
assume  x in dom h ; ::_thesis: h . x in dom f
then  h . x in rng h by FUNCT_1:3;
hence  h . x in dom f by A1; ::_thesis: verum
end;
then A18: not x in dom h by A17, FUNCT_1:11;
 x in dom (f * g) by A15, A17, XBOOLE_0:def_3;
then A19: x in dom g by FUNCT_1:11;
hence (f * (g +* h)) . x = f . ((g +* h) . x) by A3, FUNCT_1:13
.= f . (g . x) by A18, FUNCT_4:11
.= (f * g) . x by A19, FUNCT_1:13 ;
::_thesis: verum
end;
hence  f * (g +* h) = (f * g) +* (f * h) by A4, FUNCT_4:def_1; ::_thesis: verum
end;

theorem Th10: :: FUNCT_7:10
for f, g, h being Function holds (g +* h) * f = (g * f) +* (h * f)
proof
let f, g, h be Function; ::_thesis: (g +* h) * f = (g * f) +* (h * f)
A1: dom ((g +* h) * f) = (dom (g * f)) \/ (dom (h * f))
proof
thus  dom ((g +* h) * f) c= (dom (g * f)) \/ (dom (h * f)) :: according to XBOOLE_0:def_10 ::_thesis: (dom (g * f)) \/ (dom (h * f)) c= dom ((g +* h) * f)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom ((g +* h) * f) or x in (dom (g * f)) \/ (dom (h * f)) )
assume A2: x in dom ((g +* h) * f) ; ::_thesis: x in (dom (g * f)) \/ (dom (h * f))
then  f . x in dom (g +* h) by FUNCT_1:11;
then  f . x in (dom g) \/ (dom h) by FUNCT_4:def_1;
then A3: ( f . x in dom g or f . x in dom h ) by XBOOLE_0:def_3;
 x in dom f by A2, FUNCT_1:11;
then  ( x in dom (g * f) or x in dom (h * f) ) by A3, FUNCT_1:11;
hence  x in (dom (g * f)) \/ (dom (h * f)) by XBOOLE_0:def_3; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (dom (g * f)) \/ (dom (h * f)) or x in dom ((g +* h) * f) )
assume  x in (dom (g * f)) \/ (dom (h * f)) ; ::_thesis: x in dom ((g +* h) * f)
then A4: ( x in dom (g * f) or x in dom (h * f) ) by XBOOLE_0:def_3;
then  ( f . x in dom g or f . x in dom h ) by FUNCT_1:11;
then  f . x in (dom g) \/ (dom h) by XBOOLE_0:def_3;
then A5: f . x in dom (g +* h) by FUNCT_4:def_1;
 x in dom f by A4, FUNCT_1:11;
hence  x in dom ((g +* h) * f) by A5, FUNCT_1:11; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_x_in_(dom_(g_*_f))_\/_(dom_(h_*_f))_holds_
(_(_x_in_dom_(h_*_f)_implies_((g_+*_h)_*_f)_._x_=_(h_*_f)_._x_)_&_(_not_x_in_dom_(h_*_f)_implies_((g_+*_h)_*_f)_._x_=_(g_*_f)_._x_)_)
let x be set ; ::_thesis: ( x in (dom (g * f)) \/ (dom (h * f)) implies ( ( x in dom (h * f) implies ((g +* h) * f) . x = (h * f) . x ) & ( not x in dom (h * f) implies ((g +* h) * f) . x = (g * f) . x ) ) )
assume  x in (dom (g * f)) \/ (dom (h * f)) ; ::_thesis: ( ( x in dom (h * f) implies ((g +* h) * f) . x = (h * f) . x ) & ( not x in dom (h * f) implies ((g +* h) * f) . x = (g * f) . x ) )
then  ( x in dom (g * f) or x in dom (h * f) ) by XBOOLE_0:def_3;
then A6: x in dom f by FUNCT_1:11;
hereby  ::_thesis: ( not x in dom (h * f) implies ((g +* h) * f) . x = (g * f) . x )
assume A7: x in dom (h * f) ; ::_thesis: ((g +* h) * f) . x = (h * f) . x
then A8: f . x in dom h by FUNCT_1:11;
thus ((g +* h) * f) . x = (g +* h) . (f . x) by A6, FUNCT_1:13
.= h . (f . x) by A8, FUNCT_4:13
.= (h * f) . x by A7, FUNCT_1:12 ; ::_thesis: verum
end;
assume  not x in dom (h * f) ; ::_thesis: ((g +* h) * f) . x = (g * f) . x
then A9: not f . x in dom h by A6, FUNCT_1:11;
thus ((g +* h) * f) . x = (g +* h) . (f . x) by A6, FUNCT_1:13
.= g . (f . x) by A9, FUNCT_4:11
.= (g * f) . x by A6, FUNCT_1:13 ; ::_thesis: verum
end;
hence  (g +* h) * f = (g * f) +* (h * f) by A1, FUNCT_4:def_1; ::_thesis: verum
end;

theorem :: FUNCT_7:11
for f, g, h being Function st rng f misses dom g holds
(h +* g) * f = h * f
proof
let f, g, h be Function; ::_thesis: ( rng f misses dom g implies (h +* g) * f = h * f )
assume A1: rng f misses dom g ; ::_thesis: (h +* g) * f = h * f
thus (h +* g) * f = (h * f) +* (g * f) by Th10
.= (h * f) +* {} by A1, RELAT_1:44
.= h * f ; ::_thesis: verum
end;

theorem Th12: :: FUNCT_7:12
for A, B, y being set st A meets rng ((id B) +* (A --> y)) holds
y in A
proof
let A, B be set ; ::_thesis: for y being set st A meets rng ((id B) +* (A --> y)) holds
y in A
let y be set ; ::_thesis: ( A meets rng ((id B) +* (A --> y)) implies y in A )
assume  A meets rng ((id B) +* (A --> y)) ; ::_thesis: y in A
then consider x being set  such that
A1: x in A and
A2: x in rng ((id B) +* (A --> y)) by XBOOLE_0:3;
consider z being set  such that
A3: z in dom ((id B) +* (A --> y)) and
A4: ((id B) +* (A --> y)) . z = x by A2, FUNCT_1:def_3;
A5: z in (dom (id B)) \/ (dom (A --> y)) by A3, FUNCT_4:def_1;
percases ( z in dom (A --> y) or not z in dom (A --> y) ) ;
supposeA6: z in dom (A --> y) ; ::_thesis: y in A
then y = (A --> y) . z by FUNCOP_1:7
.= x by A4, A6, FUNCT_4:13 ;
hence  y in A by A1; ::_thesis: verum
end;
supposeA7: not z in dom (A --> y) ; ::_thesis: y in A
then A8: z in dom (id B) by A5, XBOOLE_0:def_3;
x = (id B) . z by A4, A7, FUNCT_4:11
.= z by A8, FUNCT_1:18 ;
hence  y in A by A1, A7, FUNCOP_1:13; ::_thesis: verum
end;
end;
end;

theorem :: FUNCT_7:13
for x, y, A being set st x <> y holds
not x in rng ((id A) +* (x .--> y))
proof
let x, y be set ; ::_thesis: for A being set st x <> y holds
not x in rng ((id A) +* (x .--> y))
let A be set ; ::_thesis: ( x <> y implies not x in rng ((id A) +* (x .--> y)) )
assume  x <> y ; ::_thesis: not x in rng ((id A) +* (x .--> y))
then  not y in {x} by TARSKI:def_1;
then  {x} misses rng ((id A) +* ({x} --> y)) by Th12;
hence  not x in rng ((id A) +* (x .--> y)) by ZFMISC_1:48; ::_thesis: verum
end;

theorem :: FUNCT_7:14
for X, a being set
for f being Function st dom f = X \/ {a} holds
f = (f | X) +* (a .--> (f . a))
proof
let X, a be set ; ::_thesis: for f being Function st dom f = X \/ {a} holds
f = (f | X) +* (a .--> (f . a))
let f be Function; ::_thesis: ( dom f = X \/ {a} implies f = (f | X) +* (a .--> (f . a)) )
assume A1: dom f = X \/ {a} ; ::_thesis: f = (f | X) +* (a .--> (f . a))
 a in {a} by TARSKI:def_1;
then A2: a in dom f by A1, XBOOLE_0:def_3;
thus f = (f | X) +* (f | {a}) by A1, FUNCT_4:70
.= (f | X) +* (a .--> (f . a)) by A2, Th6 ; ::_thesis: verum
end;

theorem :: FUNCT_7:15
for f being Function
for X, y, z being set holds (f +* (X --> y)) +* (X --> z) = f +* (X --> z)
proof
let f be Function; ::_thesis: for X, y, z being set holds (f +* (X --> y)) +* (X --> z) = f +* (X --> z)
let X, y, z be set ; ::_thesis: (f +* (X --> y)) +* (X --> z) = f +* (X --> z)
A1: dom (X --> z) = X by FUNCOP_1:13;
A2: dom (X --> y) = X by FUNCOP_1:13;
then  dom (f +* (X --> y)) = (dom f) \/ X by FUNCT_4:def_1;
then A3: dom ((f +* (X --> y)) +* (X --> z)) = ((dom f) \/ X) \/ X by A1, FUNCT_4:def_1
.= (dom f) \/ (X \/ X) by XBOOLE_1:4
.= (dom f) \/ X ;
A4: now__::_thesis:_for_x_being_set_st_x_in_(dom_f)_\/_X_holds_
((f_+*_(X_-->_y))_+*_(X_-->_z))_._x_=_(f_+*_(X_-->_z))_._x
let x be set ; ::_thesis: ( x in (dom f) \/ X implies ((f +* (X --> y)) +* (X --> z)) . b1 = (f +* (X --> z)) . b1 )
assume  x in (dom f) \/ X ; ::_thesis: ((f +* (X --> y)) +* (X --> z)) . b1 = (f +* (X --> z)) . b1
percases ( x in X or not x in X ) ;
supposeA5: x in X ; ::_thesis: ((f +* (X --> y)) +* (X --> z)) . b1 = (f +* (X --> z)) . b1
then  ((f +* (X --> y)) +* (X --> z)) . x = (X --> z) . x by A1, FUNCT_4:13;
hence  ((f +* (X --> y)) +* (X --> z)) . x = (f +* (X --> z)) . x by A1, A5, FUNCT_4:13; ::_thesis: verum
end;
supposeA6: not x in X ; ::_thesis: ((f +* (X --> y)) +* (X --> z)) . b1 = (f +* (X --> z)) . b1
then  ((f +* (X --> y)) +* (X --> z)) . x = (f +* (X --> y)) . x by A1, FUNCT_4:11;
then  ((f +* (X --> y)) +* (X --> z)) . x = f . x by A2, A6, FUNCT_4:11;
hence  ((f +* (X --> y)) +* (X --> z)) . x = (f +* (X --> z)) . x by A1, A6, FUNCT_4:11; ::_thesis: verum
end;
end;
end;
 dom (f +* (X --> z)) = (dom f) \/ X by A1, FUNCT_4:def_1;
hence  (f +* (X --> y)) +* (X --> z) = f +* (X --> z) by A3, A4, FUNCT_1:2; ::_thesis: verum
end;

theorem :: FUNCT_7:16
INT <> INT *
proof
now__::_thesis:_not_1_in_INT_*
A1: {} in 1 by CARD_1:49, TARSKI:def_1;
assume  1 in INT * ; ::_thesis: contradiction
then  ex x, y being set st {} = [x,y] by A1, RELAT_1:def_1;
hence  contradiction ; ::_thesis: verum
end;
hence  INT <> INT * by INT_1:def_2; ::_thesis: verum
end;

theorem :: FUNCT_7:17
{} * = {{}}
proof
thus  {} * c= {{}} :: according to XBOOLE_0:def_10 ::_thesis: {{}} c= {} *
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {} * or x in {{}} )
assume  x in {} * ; ::_thesis: x in {{}}
then reconsider f = x as FinSequence of {} by FINSEQ_1:def_11;
now__::_thesis:_not_x_<>_{}
assume  x <> {} ; ::_thesis: contradiction
then  ex x being set st x in dom f ;
hence  contradiction ; ::_thesis: verum
end;
hence  x in {{}} by ZFMISC_1:31; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {{}} or x in {} * )
assume  x in {{}} ; ::_thesis: x in {} *
then A1: x = {} by TARSKI:def_1;
 rng {} = {} ;
then  x is FinSequence of {} by A1, FINSEQ_1:def_4;
hence  x in {} * by FINSEQ_1:def_11; ::_thesis: verum
end;

theorem Th18: :: FUNCT_7:18
for x, A being set holds
( <*x*> in A * iff x in A )
proof
let x, A be set ; ::_thesis: ( <*x*> in A * iff x in A )
 rng <*x*> = {x} by FINSEQ_1:38;
then  ( {x} c= A iff <*x*> is FinSequence of A ) by FINSEQ_1:def_4;
hence  ( <*x*> in A * iff x in A ) by FINSEQ_1:def_11, ZFMISC_1:31; ::_thesis: verum
end;

theorem :: FUNCT_7:19
for A, B being set st A * c= B * holds
A c= B
proof
let A, B be set ; ::_thesis: ( A * c= B * implies A c= B )
assume A1: A * c= B * ; ::_thesis: A c= B
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in B )
assume  x in A ; ::_thesis: x in B
then  <*x*> in A * by Th18;
hence  x in B by A1, Th18; ::_thesis: verum
end;

theorem :: FUNCT_7:20
for A being Subset of NAT st ( for n, m being Element of NAT st n in A & m < n holds
m in A ) holds
A is Cardinal
proof
let A be Subset of NAT; ::_thesis: ( ( for n, m being Element of NAT st n in A & m < n holds
m in A ) implies A is Cardinal )
assume A1: for n, m being Element of NAT st n in A & m < n holds
m in A ; ::_thesis: A is Cardinal
percases ( A = {} or ( A <> {} & ex m being Nat st
for n being Nat st n in A holds
n <= m ) or for m being Nat ex n being Nat st
( n in A & n > m ) ) ;
suppose A = {} ; ::_thesis: A is Cardinal
hence  A is Cardinal ; ::_thesis: verum
end;
supposethat A2: A <> {} and
A3: ex m being Nat st
for n being Nat st n in A holds
n <= m ; ::_thesis: A is Cardinal
defpred S1[ set ] means $1 in A;
consider M being Nat such that
A4: for n being Nat st S1[n] holds
n <= M by A3;
 ex m being Element of NAT st S1[m] by A2, SUBSET_1:4;
then A5: ex m being Nat st S1[m] ;
consider m being Nat such that
A6: S1[m] and
A7: for n being Nat st S1[n] holds
n <= m from NAT_1:sch_6(A4, A5);
 A =  {  l where l is Element of NAT : l < m + 1  }
proof
thus  A c=  {  l where l is Element of NAT : l < m + 1  }  :: according to XBOOLE_0:def_10 ::_thesis:  {  l where l is Element of NAT : l < m + 1  }  c= A
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in  {  l where l is Element of NAT : l < m + 1  }  )
assume A8: x in A ; ::_thesis: x in  {  l where l is Element of NAT : l < m + 1  }
then reconsider l = x as Element of NAT ;
 l <= m by A7, A8;
then  l < m + 1 by NAT_1:13;
hence  x in  {  l where l is Element of NAT : l < m + 1  }  ; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  l where l is Element of NAT : l < m + 1  }  or x in A )
assume  x in  {  l where l is Element of NAT : l < m + 1  }  ; ::_thesis: x in A
then consider l being Element of NAT  such that
A9: x = l and
A10: l < m + 1 ;
 l <= m by A10, NAT_1:13;
then  ( l < m or l = m ) by XXREAL_0:1;
hence  x in A by A1, A6, A9; ::_thesis: verum
end;
hence  A is Cardinal by AXIOMS:4; ::_thesis: verum
end;
supposeA11: for m being Nat ex n being Nat st
( n in A & n > m ) ; ::_thesis: A is Cardinal
 NAT c= A
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in NAT or x in A )
assume  x in NAT ; ::_thesis: x in A
then reconsider m = x as Element of NAT ;
 ex n being Nat st
( n in A & n > m ) by A11;
hence  x in A by A1; ::_thesis: verum
end;
hence  A is Cardinal by XBOOLE_0:def_10; ::_thesis: verum
end;
end;
end;

theorem :: FUNCT_7:21
for A being finite set
for X being non empty Subset-Family of A ex C being Element of X st
for B being Element of X st B c= C holds
B = C
proof
let A be finite set ; ::_thesis: for X being non empty Subset-Family of A ex C being Element of X st
for B being Element of X st B c= C holds
B = C
let X be non empty Subset-Family of A; ::_thesis: ex C being Element of X st
for B being Element of X st B c= C holds
B = C
reconsider D = COMPLEMENT X as non empty Subset-Family of A by SETFAM_1:32;
consider x being set  such that
A1: x in D and
A2: for B being set st B in D & x c= B holds
B = x by FINSET_1:6;
reconsider x = x as Subset of A by A1;
reconsider C = x ` as Element of X by A1, SETFAM_1:def_7;
take  C ; ::_thesis: for B being Element of X st B c= C holds
B = C
let B be Element of X; ::_thesis: ( B c= C implies B = C )
assume A3: B c= C ; ::_thesis: B = C
 (B `) ` = B ;
then  B ` in D by SETFAM_1:def_7;
then  B ` = x by A2, A3, SUBSET_1:16;
hence  B = C ; ::_thesis: verum
end;

theorem Th22: :: FUNCT_7:22
for p, q being FinSequence st len p = (len q) + 1 holds
for i being Element of NAT holds
( i in dom q iff ( i in dom p & i + 1 in dom p ) )
proof
let p, q be FinSequence; ::_thesis: ( len p = (len q) + 1 implies for i being Element of NAT holds
( i in dom q iff ( i in dom p & i + 1 in dom p ) ) )
assume A1: len p = (len q) + 1 ; ::_thesis: for i being Element of NAT holds
( i in dom q iff ( i in dom p & i + 1 in dom p ) )
let i be Element of NAT ; ::_thesis: ( i in dom q iff ( i in dom p & i + 1 in dom p ) )
hereby  ::_thesis: ( i in dom p & i + 1 in dom p implies i in dom q )
assume A2: i in dom q ; ::_thesis: ( i in dom p & i + 1 in dom p )
then A3: i >= 1 by FINSEQ_3:25;
A4: i <= len q by A2, FINSEQ_3:25;
 len q <= len p by A1, NAT_1:11;
then A5: ( i + 1 >= 1 & i <= len p ) by A4, NAT_1:11, XXREAL_0:2;
 i + 1 <= len p by A1, A4, XREAL_1:6;
hence  ( i in dom p & i + 1 in dom p ) by A3, A5, FINSEQ_3:25; ::_thesis: verum
end;
assume that
A6: i in dom p and
A7: i + 1 in dom p ; ::_thesis: i in dom q
 i + 1 <= len p by A7, FINSEQ_3:25;
then A8: i <= len q by A1, XREAL_1:6;
 i >= 1 by A6, FINSEQ_3:25;
hence  i in dom q by A8, FINSEQ_3:25; ::_thesis: verum
end;

registration
cluster non empty Relation-like non-empty NAT -defined Function-like Function-yielding finite FinSequence-like FinSubsequence-like for set ;
existence
 ex b1 being FinSequence st
( b1 is Function-yielding & not b1 is empty & b1 is non-empty )
proof
take p = <*<*0*>*>; ::_thesis: ( p is Function-yielding & not p is empty & p is non-empty )
A1: ( dom p = {1} & p . 1 = <*0*> ) by FINSEQ_1:2, FINSEQ_1:38, FINSEQ_1:40;
thus  p is Function-yielding ::_thesis: ( not p is empty & p is non-empty )
proof
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 p or p . x is set )
assume  x in dom p ; ::_thesis: p . x is set
hence  p . x is set by A1, TARSKI:def_1; ::_thesis: verum
end;
thus  not p is empty ; ::_thesis: p is non-empty
let x be set ; :: according to FUNCT_1:def_9 ::_thesis: ( not x in proj1 p or not p . x is empty )
assume  x in dom p ; ::_thesis: not p . x is empty
hence  not p . x is empty by A1, TARSKI:def_1; ::_thesis: verum
end;
end;

registration
cluster empty Relation-like Function-like  ->  Function-yielding for set ;
coherence
 for b1 being Function st b1 is empty holds
b1 is Function-yielding ;
let f be Function;
cluster<*f*> ->  Function-yielding ;
coherence
 <*f*> is Function-yielding
proof
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 <*f*> or <*f*> . x is set )
assume  x in dom <*f*> ; ::_thesis: <*f*> . x is set
then  x in {1} by FINSEQ_1:2, FINSEQ_1:38;
then  x = 1 by TARSKI:def_1;
hence  <*f*> . x is set by FINSEQ_1:40; ::_thesis: verum
end;
let g be Function;
cluster<*f,g*> ->  Function-yielding ;
coherence
 <*f,g*> is Function-yielding
proof
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 <*f,g*> or <*f,g*> . x is set )
assume  x in dom <*f,g*> ; ::_thesis: <*f,g*> . x is set
then  x in {1,2} by FINSEQ_1:2, FINSEQ_1:89;
then  ( x = 1 or x = 2 ) by TARSKI:def_2;
hence  <*f,g*> . x is set by FINSEQ_1:44; ::_thesis: verum
end;
let h be Function;
cluster<*f,g,h*> ->  Function-yielding ;
coherence
 <*f,g,h*> is Function-yielding
proof
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 <*f,g,h*> or <*f,g,h*> . x is set )
assume  x in dom <*f,g,h*> ; ::_thesis: <*f,g,h*> . x is set
then  x in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;
then  ( x = 1 or x = 2 or x = 3 ) by ENUMSET1:def_1;
hence  <*f,g,h*> . x is set by FINSEQ_1:45; ::_thesis: verum
end;
end;

registration
let n be Element of NAT ;
let f be Function;
clustern |-> f ->  Function-yielding ;
coherence
 n |-> f is Function-yielding ;
end;

registration
let p, q be Function-yielding FinSequence;
clusterp ^ q ->  Function-yielding ;
coherence
 p ^ q is Function-yielding
proof
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 (p ^ q) or (p ^ q) . x is set )
assume A1: x in dom (p ^ q) ; ::_thesis: (p ^ q) . x is set
then reconsider i = x as Element of NAT ;
percases ( i in dom p or not i in dom p ) ;
suppose i in dom p ; ::_thesis: (p ^ q) . x is set
then  p . i = (p ^ q) . i by FINSEQ_1:def_7;
hence  (p ^ q) . x is set ; ::_thesis: verum
end;
suppose not i in dom p ; ::_thesis: (p ^ q) . x is set
then consider j being Nat such that
A2: ( j in dom q & i = (len p) + j ) by A1, FINSEQ_1:25;
 q . j = (p ^ q) . i by A2, FINSEQ_1:def_7;
hence  (p ^ q) . x is set ; ::_thesis: verum
end;
end;
end;
end;

theorem Th23: :: FUNCT_7:23
for p, q being FinSequence st p ^ q is Function-yielding holds
( p is Function-yielding & q is Function-yielding )
proof
let p, q be FinSequence; ::_thesis: ( p ^ q is Function-yielding implies ( p is Function-yielding & q is Function-yielding ) )
assume A1: for x being set st x in dom (p ^ q) holds
(p ^ q) . x is Function ; :: according to FUNCOP_1:def_6 ::_thesis: ( p is Function-yielding & q is Function-yielding )
hereby  :: according to FUNCOP_1:def_6 ::_thesis: q is Function-yielding
let x be set ; ::_thesis: ( x in dom p implies p . x is Function )
assume A2: x in dom p ; ::_thesis: p . x is Function
then  (p ^ q) . x = p . x by FINSEQ_1:def_7;
hence  p . x is Function by A1, A2, FINSEQ_3:22; ::_thesis: verum
end;
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 q or q . x is set )
assume A3: x in dom q ; ::_thesis: q . x is set
then reconsider i = x as Element of NAT ;
 (p ^ q) . ((len p) + i) = q . x by A3, FINSEQ_1:def_7;
hence  q . x is set by A1, A3, FINSEQ_1:28; ::_thesis: verum
end;

begin

scheme :: FUNCT_7:sch 1
Kappa2D{ F1() ->  non empty set , F2() ->  non empty set , F3() ->  non empty set , F4( Element of F1(), Element of F2()) ->  set } :
 ex f being Function of [:F1(),F2():],F3() st
for x being Element of F1()
for y being Element of F2() holds f . (x,y) = F4(x,y)
provided
A1: for x being Element of F1()
for y being Element of F2() holds F4(x,y) in F3()
proof
deffunc H1( Element of [:F1(),F2():]) ->  set = F4(($1 `1),($1 `2));
A2: for p being Element of [:F1(),F2():] holds H1(p) in F3() by A1;
consider f being Function of [:F1(),F2():],F3() such that
A3: for p being Element of [:F1(),F2():] holds f . p = H1(p) from FUNCT_2:sch_8(A2);
take  f ; ::_thesis: for x being Element of F1()
for y being Element of F2() holds f . (x,y) = F4(x,y)
let x be Element of F1(); ::_thesis: for y being Element of F2() holds f . (x,y) = F4(x,y)
let y be Element of F2(); ::_thesis: f . (x,y) = F4(x,y)
 ( [x,y] `1 = x & [x,y] `2 = y ) ;
hence  f . (x,y) = F4(x,y) by A3; ::_thesis: verum
end;

scheme :: FUNCT_7:sch 2
FinMono{ F1() ->  set , F2() ->  non empty set , F3( set ) ->  set , F4( set ) ->  set } :
  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  is finite
provided
A1: F1() is finite and
A2: for d1, d2 being Element of F2() st F4(d1) = F4(d2) holds
d1 = d2
proof
percases ( F1() = {} or F1() <> {} ) ;
supposeA3: F1() = {} ; ::_thesis:  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  is finite
now__::_thesis:_not__{__F3(d)_where_d_is_Element_of_F2()_:_F4(d)_in_F1()__}__<>_{}
set a = the Element of  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  ;
assume   {  F3(d) where d is Element of F2() : F4(d) in F1()  }  <> {} ; ::_thesis: contradiction
then  the Element of  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  in  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  ;
then  ex d being Element of F2() st
( the Element of  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  = F3(d) & F4(d) in F1() ) ;
hence  contradiction by A3; ::_thesis: verum
end;
hence   {  F3(d) where d is Element of F2() : F4(d) in F1()  }  is finite ; ::_thesis: verum
end;
suppose F1() <> {} ; ::_thesis:  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  is finite
then reconsider D = F1() as non empty set ;
defpred S1[ set ] means  ex d being Element of F2() st $1 = F4(d);
set B =  {  d where d is Element of F2() : F4(d) in D  }  ;
set C =  {  a where a is Element of D : S1[a]  }  ;
A4:  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  =  {  F3(d) where d is Element of F2() : d in  {  d where d is Element of F2() : F4(d) in D  }   }
proof
thus   {  F3(d) where d is Element of F2() : F4(d) in F1()  }  c=  {  F3(d) where d is Element of F2() : d in  {  d where d is Element of F2() : F4(d) in D  }   }  :: according to XBOOLE_0:def_10 ::_thesis:  {  F3(d) where d is Element of F2() : d in  {  d where d is Element of F2() : F4(d) in D  }   }  c=  {  F3(d) where d is Element of F2() : F4(d) in F1()  }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  or x in  {  F3(d) where d is Element of F2() : d in  {  d where d is Element of F2() : F4(d) in D  }   }  )
assume  x in  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  ; ::_thesis: x in  {  F3(d) where d is Element of F2() : d in  {  d where d is Element of F2() : F4(d) in D  }   }
then consider d9 being Element of F2() such that
A5: x = F3(d9) and
A6: F4(d9) in F1() ;
 d9 in  {  d where d is Element of F2() : F4(d) in D  }  by A6;
hence  x in  {  F3(d) where d is Element of F2() : d in  {  d where d is Element of F2() : F4(d) in D  }   }  by A5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  F3(d) where d is Element of F2() : d in  {  d where d is Element of F2() : F4(d) in D  }   }  or x in  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  )
assume  x in  {  F3(d) where d is Element of F2() : d in  {  d where d is Element of F2() : F4(d) in D  }   }  ; ::_thesis: x in  {  F3(d) where d is Element of F2() : F4(d) in F1()  }
then consider d1 being Element of F2() such that
A7: x = F3(d1) and
A8: d1 in  {  d where d is Element of F2() : F4(d) in D  }  ;
 ex d3 being Element of F2() st
( d3 = d1 & F4(d3) in D ) by A8;
hence  x in  {  F3(d) where d is Element of F2() : F4(d) in F1()  }  by A7; ::_thesis: verum
end;
A9:  {  a where a is Element of D : S1[a]  }  , {  d where d is Element of F2() : F4(d) in D  }  are_equipotent
proof
take Z =  {  [F4(d),d] where d is Element of F2() : verum  }  ; :: according to TARSKI:def_6 ::_thesis: ( ( for b1 being set holds
( not b1 in  {  a where a is Element of D : S1[a]  }  or ex b2 being set st
( b2 in  {  d where d is Element of F2() : F4(d) in D  }  & [b1,b2] in Z ) ) ) & ( for b1 being set holds
( not b1 in  {  d where d is Element of F2() : F4(d) in D  }  or ex b2 being set st
( b2 in  {  a where a is Element of D : S1[a]  }  & [b2,b1] in Z ) ) ) & ( for b1, b2, b3, b4 being set holds
( not [b1,b2] in Z or not [b3,b4] in Z or ( ( not b1 = b3 or b2 = b4 ) & ( not b2 = b4 or b1 = b3 ) ) ) ) )
hereby  ::_thesis: ( ( for b1 being set holds
( not b1 in  {  d where d is Element of F2() : F4(d) in D  }  or ex b2 being set st
( b2 in  {  a where a is Element of D : S1[a]  }  & [b2,b1] in Z ) ) ) & ( for b1, b2, b3, b4 being set holds
( not [b1,b2] in Z or not [b3,b4] in Z or ( ( not b1 = b3 or b2 = b4 ) & ( not b2 = b4 or b1 = b3 ) ) ) ) )
let x be set ; ::_thesis: ( x in  {  a where a is Element of D : S1[a]  }  implies ex d being set st
( d in  {  d where d is Element of F2() : F4(d) in D  }  & [x,d] in Z ) )
assume  x in  {  a where a is Element of D : S1[a]  }  ; ::_thesis: ex d being set st
( d in  {  d where d is Element of F2() : F4(d) in D  }  & [x,d] in Z )
then consider a being Element of D such that
A10: a = x and
A11: ex d being Element of F2() st a = F4(d) ;
consider d being Element of F2() such that
A12: a = F4(d) by A11;
reconsider d = d as set ;
take d = d; ::_thesis: ( d in  {  d where d is Element of F2() : F4(d) in D  }  & [x,d] in Z )
thus  ( d in  {  d where d is Element of F2() : F4(d) in D  }  & [x,d] in Z ) by A10, A12; ::_thesis: verum
end;
hereby  ::_thesis: for b1, b2, b3, b4 being set holds
( not [b1,b2] in Z or not [b3,b4] in Z or ( ( not b1 = b3 or b2 = b4 ) & ( not b2 = b4 or b1 = b3 ) ) )
let y be set ; ::_thesis: ( y in  {  d where d is Element of F2() : F4(d) in D  }  implies ex x being set st
( x in  {  a where a is Element of D : S1[a]  }  & [x,y] in Z ) )
assume  y in  {  d where d is Element of F2() : F4(d) in D  }  ; ::_thesis: ex x being set st
( x in  {  a where a is Element of D : S1[a]  }  & [x,y] in Z )
then consider d being Element of F2() such that
A13: ( d = y & F4(d) in D ) ;
reconsider x = F4(d) as set ;
take x = x; ::_thesis: ( x in  {  a where a is Element of D : S1[a]  }  & [x,y] in Z )
thus  ( x in  {  a where a is Element of D : S1[a]  }  & [x,y] in Z ) by A13; ::_thesis: verum
end;
let x, y, z, u be set ; ::_thesis: ( not [x,y] in Z or not [z,u] in Z or ( ( not x = z or y = u ) & ( not y = u or x = z ) ) )
assume  [x,y] in Z ; ::_thesis: ( not [z,u] in Z or ( ( not x = z or y = u ) & ( not y = u or x = z ) ) )
then consider d1 being Element of F2() such that
A14: [x,y] = [F4(d1),d1] ;
assume  [z,u] in Z ; ::_thesis: ( ( not x = z or y = u ) & ( not y = u or x = z ) )
then consider d2 being Element of F2() such that
A15: [z,u] = [F4(d2),d2] ;
A16: ( z = F4(d2) & u = d2 ) by A15, XTUPLE_0:1;
 ( x = F4(d1) & y = d1 ) by A14, XTUPLE_0:1;
hence  ( ( not x = z or y = u ) & ( not y = u or x = z ) ) by A2, A16; ::_thesis: verum
end;
  {  a where a is Element of D : S1[a]  }  is Subset of D from DOMAIN_1:sch_7();
then   {  a where a is Element of D : S1[a]  }  is finite by A1, FINSET_1:1;
then A17:  {  d where d is Element of F2() : F4(d) in D  }  is finite by A9, CARD_1:38;
  {  F3(d) where d is Element of F2() : d in  {  d where d is Element of F2() : F4(d) in D  }   }  is finite from FRAENKEL:sch_21(A17);
hence   {  F3(d) where d is Element of F2() : F4(d) in F1()  }  is finite by A4; ::_thesis: verum
end;
end;
end;

scheme :: FUNCT_7:sch 3
CardMono{ F1() ->  set , F2() ->  non empty set , F3( set ) ->  set } :
 F1(), {  d where d is Element of F2() : F3(d) in F1()  }  are_equipotent
provided
A1: for x being set st x in F1() holds
ex d being Element of F2() st x = F3(d) and
A2: for d1, d2 being Element of F2() st F3(d1) = F3(d2) holds
d1 = d2
proof
set D =  {  d where d is Element of F2() : F3(d) in F1()  }  ;
percases ( F1() = {} or F1() <> {} ) ;
supposeA3: F1() = {} ; ::_thesis: F1(), {  d where d is Element of F2() : F3(d) in F1()  }  are_equipotent
now__::_thesis:_not__{__d_where_d_is_Element_of_F2()_:_F3(d)_in_F1()__}__<>_{}
set a = the Element of  {  d where d is Element of F2() : F3(d) in F1()  }  ;
assume   {  d where d is Element of F2() : F3(d) in F1()  }  <> {} ; ::_thesis: contradiction
then  the Element of  {  d where d is Element of F2() : F3(d) in F1()  }  in  {  d where d is Element of F2() : F3(d) in F1()  }  ;
then  ex d being Element of F2() st
( the Element of  {  d where d is Element of F2() : F3(d) in F1()  }  = d & F3(d) in F1() ) ;
hence  contradiction by A3; ::_thesis: verum
end;
hence  F1(), {  d where d is Element of F2() : F3(d) in F1()  }  are_equipotent by A3; ::_thesis: verum
end;
suppose F1() <> {} ; ::_thesis: F1(), {  d where d is Element of F2() : F3(d) in F1()  }  are_equipotent
then reconsider A = F1() as non empty set ;
 A, {  d where d is Element of F2() : F3(d) in F1()  }  are_equipotent
proof
take Z =  {  [F3(d),d] where d is Element of F2() : verum  }  ; :: according to TARSKI:def_6 ::_thesis: ( ( for b1 being set holds
( not b1 in A or ex b2 being set st
( b2 in  {  d where d is Element of F2() : F3(d) in F1()  }  & [b1,b2] in Z ) ) ) & ( for b1 being set holds
( not b1 in  {  d where d is Element of F2() : F3(d) in F1()  }  or ex b2 being set st
( b2 in A & [b2,b1] in Z ) ) ) & ( for b1, b2, b3, b4 being set holds
( not [b1,b2] in Z or not [b3,b4] in Z or ( ( not b1 = b3 or b2 = b4 ) & ( not b2 = b4 or b1 = b3 ) ) ) ) )
hereby  ::_thesis: ( ( for b1 being set holds
( not b1 in  {  d where d is Element of F2() : F3(d) in F1()  }  or ex b2 being set st
( b2 in A & [b2,b1] in Z ) ) ) & ( for b1, b2, b3, b4 being set holds
( not [b1,b2] in Z or not [b3,b4] in Z or ( ( not b1 = b3 or b2 = b4 ) & ( not b2 = b4 or b1 = b3 ) ) ) ) )
let x be set ; ::_thesis: ( x in A implies ex d being set st
( d in  {  d where d is Element of F2() : F3(d) in F1()  }  & [x,d] in Z ) )
assume A4: x in A ; ::_thesis: ex d being set st
( d in  {  d where d is Element of F2() : F3(d) in F1()  }  & [x,d] in Z )
then consider d being Element of F2() such that
A5: x = F3(d) by A1;
reconsider d = d as set ;
take d = d; ::_thesis: ( d in  {  d where d is Element of F2() : F3(d) in F1()  }  & [x,d] in Z )
thus  d in  {  d where d is Element of F2() : F3(d) in F1()  }  by A4, A5; ::_thesis: [x,d] in Z
thus  [x,d] in Z by A5; ::_thesis: verum
end;
hereby  ::_thesis: for b1, b2, b3, b4 being set holds
( not [b1,b2] in Z or not [b3,b4] in Z or ( ( not b1 = b3 or b2 = b4 ) & ( not b2 = b4 or b1 = b3 ) ) )
let y be set ; ::_thesis: ( y in  {  d where d is Element of F2() : F3(d) in F1()  }  implies ex x being set st
( x in A & [x,y] in Z ) )
assume  y in  {  d where d is Element of F2() : F3(d) in F1()  }  ; ::_thesis: ex x being set st
( x in A & [x,y] in Z )
then consider d being Element of F2() such that
A6: ( d = y & F3(d) in A ) ;
reconsider x = F3(d) as set ;
take x = x; ::_thesis: ( x in A & [x,y] in Z )
thus  ( x in A & [x,y] in Z ) by A6; ::_thesis: verum
end;
let x, y, z, u be set ; ::_thesis: ( not [x,y] in Z or not [z,u] in Z or ( ( not x = z or y = u ) & ( not y = u or x = z ) ) )
assume  [x,y] in Z ; ::_thesis: ( not [z,u] in Z or ( ( not x = z or y = u ) & ( not y = u or x = z ) ) )
then consider d1 being Element of F2() such that
A7: [x,y] = [F3(d1),d1] ;
assume  [z,u] in Z ; ::_thesis: ( ( not x = z or y = u ) & ( not y = u or x = z ) )
then consider d2 being Element of F2() such that
A8: [z,u] = [F3(d2),d2] ;
A9: ( z = F3(d2) & u = d2 ) by A8, XTUPLE_0:1;
 ( x = F3(d1) & y = d1 ) by A7, XTUPLE_0:1;
hence  ( ( not x = z or y = u ) & ( not y = u or x = z ) ) by A2, A9; ::_thesis: verum
end;
hence  F1(), {  d where d is Element of F2() : F3(d) in F1()  }  are_equipotent ; ::_thesis: verum
end;
end;
end;

scheme :: FUNCT_7:sch 4
CardMono9{ F1() ->  set , F2() ->  non empty set , F3( set ) ->  set } :
 F1(), {  F3(d) where d is Element of F2() : d in F1()  }  are_equipotent
provided
A1: F1() c= F2() and
A2: for d1, d2 being Element of F2() st F3(d1) = F3(d2) holds
d1 = d2
proof
percases ( F1() = {} or F1() <> {} ) ;
supposeA3: F1() = {} ; ::_thesis: F1(), {  F3(d) where d is Element of F2() : d in F1()  }  are_equipotent
now__::_thesis:_not__{__F3(d)_where_d_is_Element_of_F2()_:_d_in_F1()__}__<>_{}
set a = the Element of  {  F3(d) where d is Element of F2() : d in F1()  }  ;
assume   {  F3(d) where d is Element of F2() : d in F1()  }  <> {} ; ::_thesis: contradiction
then  the Element of  {  F3(d) where d is Element of F2() : d in F1()  }  in  {  F3(d) where d is Element of F2() : d in F1()  }  ;
then  ex d being Element of F2() st
( the Element of  {  F3(d) where d is Element of F2() : d in F1()  }  = F3(d) & d in F1() ) ;
hence  contradiction by A3; ::_thesis: verum
end;
hence  F1(), {  F3(d) where d is Element of F2() : d in F1()  }  are_equipotent by A3; ::_thesis: verum
end;
suppose F1() <> {} ; ::_thesis: F1(), {  F3(d) where d is Element of F2() : d in F1()  }  are_equipotent
then reconsider A = F1() as non empty set ;
set D =  {  F3(d) where d is Element of F2() : d in A  }  ;
 A, {  F3(d) where d is Element of F2() : d in A  }  are_equipotent
proof
take Z =  {  [d,F3(d)] where d is Element of F2() : verum  }  ; :: according to TARSKI:def_6 ::_thesis: ( ( for b1 being set holds
( not b1 in A or ex b2 being set st
( b2 in  {  F3(d) where d is Element of F2() : d in A  }  & [b1,b2] in Z ) ) ) & ( for b1 being set holds
( not b1 in  {  F3(d) where d is Element of F2() : d in A  }  or ex b2 being set st
( b2 in A & [b2,b1] in Z ) ) ) & ( for b1, b2, b3, b4 being set holds
( not [b1,b2] in Z or not [b3,b4] in Z or ( ( not b1 = b3 or b2 = b4 ) & ( not b2 = b4 or b1 = b3 ) ) ) ) )
hereby  ::_thesis: ( ( for b1 being set holds
( not b1 in  {  F3(d) where d is Element of F2() : d in A  }  or ex b2 being set st
( b2 in A & [b2,b1] in Z ) ) ) & ( for b1, b2, b3, b4 being set holds
( not [b1,b2] in Z or not [b3,b4] in Z or ( ( not b1 = b3 or b2 = b4 ) & ( not b2 = b4 or b1 = b3 ) ) ) ) )
let x be set ; ::_thesis: ( x in A implies ex y being set st
( y in  {  F3(d) where d is Element of F2() : d in A  }  & [x,y] in Z ) )
assume A4: x in A ; ::_thesis: ex y being set st
( y in  {  F3(d) where d is Element of F2() : d in A  }  & [x,y] in Z )
then reconsider d = x as Element of F2() by A1;
take y = F3(d); ::_thesis: ( y in  {  F3(d) where d is Element of F2() : d in A  }  & [x,y] in Z )
thus  y in  {  F3(d) where d is Element of F2() : d in A  }  by A4; ::_thesis: [x,y] in Z
thus  [x,y] in Z ; ::_thesis: verum
end;
hereby  ::_thesis: for b1, b2, b3, b4 being set holds
( not [b1,b2] in Z or not [b3,b4] in Z or ( ( not b1 = b3 or b2 = b4 ) & ( not b2 = b4 or b1 = b3 ) ) )
let y be set ; ::_thesis: ( y in  {  F3(d) where d is Element of F2() : d in A  }  implies ex d being set st
( d in A & [d,y] in Z ) )
assume  y in  {  F3(d) where d is Element of F2() : d in A  }  ; ::_thesis: ex d being set st
( d in A & [d,y] in Z )
then consider d being Element of F2() such that
A5: ( F3(d) = y & d in A ) ;
reconsider d = d as set ;
take d = d; ::_thesis: ( d in A & [d,y] in Z )
thus  ( d in A & [d,y] in Z ) by A5; ::_thesis: verum
end;
let x, y, z, u be set ; ::_thesis: ( not [x,y] in Z or not [z,u] in Z or ( ( not x = z or y = u ) & ( not y = u or x = z ) ) )
assume  [x,y] in Z ; ::_thesis: ( not [z,u] in Z or ( ( not x = z or y = u ) & ( not y = u or x = z ) ) )
then consider d1 being Element of F2() such that
A6: [x,y] = [d1,F3(d1)] ;
assume  [z,u] in Z ; ::_thesis: ( ( not x = z or y = u ) & ( not y = u or x = z ) )
then consider d2 being Element of F2() such that
A7: [z,u] = [d2,F3(d2)] ;
A8: ( z = d2 & u = F3(d2) ) by A7, XTUPLE_0:1;
 ( x = d1 & y = F3(d1) ) by A6, XTUPLE_0:1;
hence  ( ( not x = z or y = u ) & ( not y = u or x = z ) ) by A2, A8; ::_thesis: verum
end;
hence  F1(), {  F3(d) where d is Element of F2() : d in F1()  }  are_equipotent ; ::_thesis: verum
end;
end;
end;

scheme :: FUNCT_7:sch 5
FuncSeqInd{ P1[ set ] } :
 for p being Function-yielding FinSequence holds P1[p]
provided
A1: P1[ {} ] and
A2: for p being Function-yielding FinSequence st P1[p] holds
for f being Function holds P1[p ^ <*f*>]
proof
defpred S1[ FinSequence] means ( $1 is Function-yielding implies P1[$1] );
A3: for p being FinSequence
for x being set st S1[p] holds
S1[p ^ <*x*>]
proof
let p be FinSequence; ::_thesis: for x being set st S1[p] holds
S1[p ^ <*x*>]
let x be set ; ::_thesis: ( S1[p] implies S1[p ^ <*x*>] )
assume that
A4: ( p is Function-yielding implies P1[p] ) and
A5: p ^ <*x*> is Function-yielding ; ::_thesis: P1[p ^ <*x*>]
A6: <*x*> . 1 = x by FINSEQ_1:40;
 ( p is Function-yielding & <*x*> is Function-yielding ) by A5, Th23;
hence  P1[p ^ <*x*>] by A2, A4, A6; ::_thesis: verum
end;
A7: S1[ {} ] by A1;
 for p being FinSequence holds S1[p] from FINSEQ_1:sch_3(A7, A3);
hence  for p being Function-yielding FinSequence holds P1[p] ; ::_thesis: verum
end;

begin

definition
let x, y be set ;
assume A1: x in y ;
func In (x,y) ->  Element of y equals :Def1: :: FUNCT_7:def 1
x;
correctness
coherence
x is Element of y;
by A1;
end;

:: deftheorem Def1 defines In FUNCT_7:def_1_:_
 for x, y being set st x in y holds
In (x,y) = x;

theorem :: FUNCT_7:24
for x, A, B being set st x in A /\ B holds
In (x,A) = In (x,B)
proof
let x, A, B be set ; ::_thesis: ( x in A /\ B implies In (x,A) = In (x,B) )
assume A1: x in A /\ B ; ::_thesis: In (x,A) = In (x,B)
then A2: x in B by XBOOLE_0:def_4;
 x in A by A1, XBOOLE_0:def_4;
hence  In (x,A) = x by Def1
.= In (x,B) by A2, Def1 ;
::_thesis: verum
end;

definition
let f, g be Function;
let A be set ;
predf,g equal_outside A means :Def2: :: FUNCT_7:def 2
f | ((dom f) \ A) = g | ((dom g) \ A);
end;

:: deftheorem Def2 defines equal_outside FUNCT_7:def_2_:_
 for f, g being Function
for A being set holds
( f,g equal_outside A iff f | ((dom f) \ A) = g | ((dom g) \ A) );

theorem Th25: :: FUNCT_7:25
for f being Function
for A being set holds f,f equal_outside A
proof
let f be Function; ::_thesis: for A being set holds f,f equal_outside A
let A be set ; ::_thesis: f,f equal_outside A
thus  f | ((dom f) \ A) = f | ((dom f) \ A) ; :: according to FUNCT_7:def_2 ::_thesis: verum
end;

theorem Th26: :: FUNCT_7:26
for f, g being Function
for A being set st f,g equal_outside A holds
g,f equal_outside A
proof
let f, g be Function; ::_thesis: for A being set st f,g equal_outside A holds
g,f equal_outside A
let A be set ; ::_thesis: ( f,g equal_outside A implies g,f equal_outside A )
assume  f | ((dom f) \ A) = g | ((dom g) \ A) ; :: according to FUNCT_7:def_2 ::_thesis: g,f equal_outside A
hence  g | ((dom g) \ A) = f | ((dom f) \ A) ; :: according to FUNCT_7:def_2 ::_thesis: verum
end;

theorem Th27: :: FUNCT_7:27
for f, g, h being Function
for A being set st f,g equal_outside A & g,h equal_outside A holds
f,h equal_outside A
proof
let f, g, h be Function; ::_thesis: for A being set st f,g equal_outside A & g,h equal_outside A holds
f,h equal_outside A
let A be set ; ::_thesis: ( f,g equal_outside A & g,h equal_outside A implies f,h equal_outside A )
assume  ( f | ((dom f) \ A) = g | ((dom g) \ A) & g | ((dom g) \ A) = h | ((dom h) \ A) ) ; :: according to FUNCT_7:def_2 ::_thesis: f,h equal_outside A
hence  f | ((dom f) \ A) = h | ((dom h) \ A) ; :: according to FUNCT_7:def_2 ::_thesis: verum
end;

theorem Th28: :: FUNCT_7:28
for f, g being Function
for A being set st f,g equal_outside A holds
(dom f) \ A = (dom g) \ A
proof
let f, g be Function; ::_thesis: for A being set st f,g equal_outside A holds
(dom f) \ A = (dom g) \ A
let A be set ; ::_thesis: ( f,g equal_outside A implies (dom f) \ A = (dom g) \ A )
assume A1: f | ((dom f) \ A) = g | ((dom g) \ A) ; :: according to FUNCT_7:def_2 ::_thesis: (dom f) \ A = (dom g) \ A
thus (dom f) \ A = (dom f) /\ ((dom f) \ A) by XBOOLE_1:28
.= dom (f | ((dom f) \ A)) by RELAT_1:61
.= (dom g) /\ ((dom g) \ A) by A1, RELAT_1:61
.= (dom g) \ A by XBOOLE_1:28 ; ::_thesis: verum
end;

theorem Th29: :: FUNCT_7:29
for f, g being Function
for A being set st dom g c= A holds
f,f +* g equal_outside A
proof
let f, g be Function; ::_thesis: for A being set st dom g c= A holds
f,f +* g equal_outside A
let A be set ; ::_thesis: ( dom g c= A implies f,f +* g equal_outside A )
assume A1: dom g c= A ; ::_thesis: f,f +* g equal_outside A
A2: (dom (f +* g)) \ A = ((dom f) \/ (dom g)) \ A by FUNCT_4:def_1
.= ((dom f) \ A) \/ ((dom g) \ A) by XBOOLE_1:42
.= ((dom f) \ A) \/ {} by A1, XBOOLE_1:37
.= (dom f) \ A ;
 (dom f) \ A misses A by XBOOLE_1:79;
then  (dom f) \ A misses dom g by A1, XBOOLE_1:63;
hence  f | ((dom f) \ A) = (f +* g) | ((dom (f +* g)) \ A) by A2, FUNCT_4:72; :: according to FUNCT_7:def_2 ::_thesis: verum
end;

definition
let f be Function;
let i, x be set ;
funcf +* (i,x) ->  Function equals :Def3: :: FUNCT_7:def 3
f +* (i .--> x) if i in dom f
otherwise f;
correctness
coherence
( ( i in dom f implies f +* (i .--> x) is Function ) & ( not i in dom f implies f is Function ) );
consistency
for b1 being Function holds verum;
;
end;

:: deftheorem Def3 defines +* FUNCT_7:def_3_:_
 for f being Function
for i, x being set holds
( ( i in dom f implies f +* (i,x) = f +* (i .--> x) ) & ( not i in dom f implies f +* (i,x) = f ) );

theorem Th30: :: FUNCT_7:30
for f being Function
for d, i being set holds dom (f +* (i,d)) = dom f
proof
let f be Function; ::_thesis: for d, i being set holds dom (f +* (i,d)) = dom f
let x, i be set ; ::_thesis: dom (f +* (i,x)) = dom f
percases ( i in dom f or not i in dom f ) ;
supposeA1: i in dom f ; ::_thesis: dom (f +* (i,x)) = dom f
then A2: {i} c= dom f by ZFMISC_1:31;
thus  dom (f +* (i,x)) = dom (f +* (i .--> x)) by A1, Def3
.= (dom f) \/ (dom (i .--> x)) by FUNCT_4:def_1
.= (dom f) \/ {i} by FUNCOP_1:13
.= dom f by A2, XBOOLE_1:12 ; ::_thesis: verum
end;
suppose not i in dom f ; ::_thesis: dom (f +* (i,x)) = dom f
hence  dom (f +* (i,x)) = dom f by Def3; ::_thesis: verum
end;
end;
end;

theorem Th31: :: FUNCT_7:31
for f being Function
for d, i being set st i in dom f holds
(f +* (i,d)) . i = d
proof
let f be Function; ::_thesis: for d, i being set st i in dom f holds
(f +* (i,d)) . i = d
let d, i be set ; ::_thesis: ( i in dom f implies (f +* (i,d)) . i = d )
 dom (i .--> d) = {i} by FUNCOP_1:13;
then A1: i in dom (i .--> d) by TARSKI:def_1;
assume  i in dom f ; ::_thesis: (f +* (i,d)) . i = d
hence (f +* (i,d)) . i = (f +* (i .--> d)) . i by Def3
.= (i .--> d) . i by A1, FUNCT_4:13
.= d by FUNCOP_1:72 ;
::_thesis: verum
end;

theorem Th32: :: FUNCT_7:32
for f being Function
for d, i, j being set st i <> j holds
(f +* (i,d)) . j = f . j
proof
let f be Function; ::_thesis: for d, i, j being set st i <> j holds
(f +* (i,d)) . j = f . j
let d, i, j be set ; ::_thesis: ( i <> j implies (f +* (i,d)) . j = f . j )
assume A1: i <> j ; ::_thesis: (f +* (i,d)) . j = f . j
A2: not j in dom (i .--> d) by A1, TARSKI:def_1;
percases ( i in dom f or not i in dom f ) ;
suppose i in dom f ; ::_thesis: (f +* (i,d)) . j = f . j
hence (f +* (i,d)) . j = (f +* (i .--> d)) . j by Def3
.= f . j by A2, FUNCT_4:11 ;
::_thesis: verum
end;
suppose not i in dom f ; ::_thesis: (f +* (i,d)) . j = f . j
hence  (f +* (i,d)) . j = f . j by Def3; ::_thesis: verum
end;
end;
end;

theorem :: FUNCT_7:33
for f being Function
for d, e, i, j being set st i <> j holds
(f +* (i,d)) +* (j,e) = (f +* (j,e)) +* (i,d)
proof
let f be Function; ::_thesis: for d, e, i, j being set st i <> j holds
(f +* (i,d)) +* (j,e) = (f +* (j,e)) +* (i,d)
let d, e, i, j be set ; ::_thesis: ( i <> j implies (f +* (i,d)) +* (j,e) = (f +* (j,e)) +* (i,d) )
assume A1: i <> j ; ::_thesis: (f +* (i,d)) +* (j,e) = (f +* (j,e)) +* (i,d)
percases ( ( i in dom f & j in dom f ) or ( i in dom f & not j in dom f ) or ( not i in dom f & j in dom f ) or ( not i in dom f & not j in dom f ) ) ;
supposethat A2: i in dom f and
A3: j in dom f ; ::_thesis: (f +* (i,d)) +* (j,e) = (f +* (j,e)) +* (i,d)
A4: i in dom (f +* (j,e)) by A2, Th30;
A5: ( dom (i .--> d) = {i} & dom (j .--> e) = {j} ) by FUNCOP_1:13;
 j in dom (f +* (i,d)) by A3, Th30;
hence (f +* (i,d)) +* (j,e) = (f +* (i,d)) +* (j .--> e) by Def3
.= (f +* (i .--> d)) +* (j .--> e) by A2, Def3
.= f +* ((i .--> d) +* (j .--> e)) by FUNCT_4:14
.= f +* ((j .--> e) +* (i .--> d)) by A1, A5, FUNCT_4:35, ZFMISC_1:11
.= (f +* (j .--> e)) +* (i .--> d) by FUNCT_4:14
.= (f +* (j,e)) +* (i .--> d) by A3, Def3
.= (f +* (j,e)) +* (i,d) by A4, Def3 ;
::_thesis: verum
end;
supposethat  i in dom f and
A6: not j in dom f ; ::_thesis: (f +* (i,d)) +* (j,e) = (f +* (j,e)) +* (i,d)
 not j in dom (f +* (i,d)) by A6, Th30;
hence (f +* (i,d)) +* (j,e) = f +* (i,d) by Def3
.= (f +* (j,e)) +* (i,d) by A6, Def3 ;
::_thesis: verum
end;
supposethat A7: not i in dom f and
 j in dom f ; ::_thesis: (f +* (i,d)) +* (j,e) = (f +* (j,e)) +* (i,d)
A8: not i in dom (f +* (j,e)) by A7, Th30;
thus (f +* (i,d)) +* (j,e) = f +* (j,e) by A7, Def3
.= (f +* (j,e)) +* (i,d) by A8, Def3 ; ::_thesis: verum
end;
supposethat A9: not i in dom f and
A10: not j in dom f ; ::_thesis: (f +* (i,d)) +* (j,e) = (f +* (j,e)) +* (i,d)
A11: not i in dom (f +* (j,e)) by A9, Th30;
 not j in dom (f +* (i,d)) by A10, Th30;
hence (f +* (i,d)) +* (j,e) = f +* (i,d) by Def3
.= f by A9, Def3
.= f +* (j,e) by A10, Def3
.= (f +* (j,e)) +* (i,d) by A11, Def3 ;
::_thesis: verum
end;
end;
end;

theorem :: FUNCT_7:34
for f being Function
for d, e, i being set holds (f +* (i,d)) +* (i,e) = f +* (i,e)
proof
let f be Function; ::_thesis: for d, e, i being set holds (f +* (i,d)) +* (i,e) = f +* (i,e)
let d, e, i be set ; ::_thesis: (f +* (i,d)) +* (i,e) = f +* (i,e)
A1: dom (i .--> d) = {i} by FUNCOP_1:13
.= dom (i .--> e) by FUNCOP_1:13 ;
percases ( i in dom f or not i in dom f ) ;
supposeA2: i in dom f ; ::_thesis: (f +* (i,d)) +* (i,e) = f +* (i,e)
then  i in dom (f +* (i,d)) by Th30;
hence (f +* (i,d)) +* (i,e) = (f +* (i,d)) +* (i .--> e) by Def3
.= (f +* (i .--> d)) +* (i .--> e) by A2, Def3
.= f +* ((i .--> d) +* (i .--> e)) by FUNCT_4:14
.= f +* (i .--> e) by A1, FUNCT_4:19
.= f +* (i,e) by A2, Def3 ;
::_thesis: verum
end;
suppose not i in dom f ; ::_thesis: (f +* (i,d)) +* (i,e) = f +* (i,e)
hence  (f +* (i,d)) +* (i,e) = f +* (i,e) by Def3; ::_thesis: verum
end;
end;
end;

theorem Th35: :: FUNCT_7:35
for f being Function
for i being set holds f +* (i,(f . i)) = f
proof
let f be Function; ::_thesis: for i being set holds f +* (i,(f . i)) = f
let i be set ; ::_thesis: f +* (i,(f . i)) = f
percases ( i in dom f or not i in dom f ) ;
supposeA1: i in dom f ; ::_thesis: f +* (i,(f . i)) = f
then A2: i .--> (f . i) = f | {i} by Th6;
thus f +* (i,(f . i)) = f +* (i .--> (f . i)) by A1, Def3
.= f by A2, FUNCT_4:75 ; ::_thesis: verum
end;
suppose not i in dom f ; ::_thesis: f +* (i,(f . i)) = f
hence  f +* (i,(f . i)) = f by Def3; ::_thesis: verum
end;
end;
end;

registration
let f be FinSequence;
let i, x be set ;
clusterf +* (i,x) ->  FinSequence-like ;
coherence
 f +* (i,x) is FinSequence-like
proof
dom (f +* (i,x)) = dom f by Th30
.= Seg (len f) by FINSEQ_1:def_3 ;
hence  f +* (i,x) is FinSequence-like by FINSEQ_1:def_2; ::_thesis: verum
end;
end;

definition
let D be set ;
let f be FinSequence of D;
let i be Element of NAT ;
let d be Element of D;
:: original: +*
redefine funcf +* (i,d) ->  FinSequence of D;
coherence
 f +* (i,d) is FinSequence of D
proof
percases ( i in dom f or not i in dom f ) ;
supposeA1: i in dom f ; ::_thesis: f +* (i,d) is FinSequence of D
then  f . i in rng f by FUNCT_1:def_3;
then  {d} c= D by ZFMISC_1:31;
then A2: ( (rng f) \/ (rng (i .--> d)) = (rng f) \/ {d} & (rng f) \/ {d} c= D ) by FUNCOP_1:8, XBOOLE_1:8;
 f +* (i,d) = f +* (i .--> d) by A1, Def3;
then  rng (f +* (i,d)) c= (rng f) \/ (rng (i .--> d)) by FUNCT_4:17;
then  rng (f +* (i,d)) c= D by A2, XBOOLE_1:1;
hence  f +* (i,d) is FinSequence of D by FINSEQ_1:def_4; ::_thesis: verum
end;
suppose not i in dom f ; ::_thesis: f +* (i,d) is FinSequence of D
hence  f +* (i,d) is FinSequence of D by Def3; ::_thesis: verum
end;
end;
end;
end;

theorem :: FUNCT_7:36
for D being non empty set
for f being FinSequence of D
for d being Element of D
for i being Element of NAT st i in dom f holds
(f +* (i,d)) /. i = d
proof
let D be non empty set ; ::_thesis: for f being FinSequence of D
for d being Element of D
for i being Element of NAT st i in dom f holds
(f +* (i,d)) /. i = d
let f be FinSequence of D; ::_thesis: for d being Element of D
for i being Element of NAT st i in dom f holds
(f +* (i,d)) /. i = d
let d be Element of D; ::_thesis: for i being Element of NAT st i in dom f holds
(f +* (i,d)) /. i = d
let i be Element of NAT ; ::_thesis: ( i in dom f implies (f +* (i,d)) /. i = d )
assume A1: i in dom f ; ::_thesis: (f +* (i,d)) /. i = d
then  i in dom (f +* (i,d)) by Th30;
hence (f +* (i,d)) /. i = (f +* (i,d)) . i by PARTFUN1:def_6
.= d by A1, Th31 ;
::_thesis: verum
end;

theorem :: FUNCT_7:37
for D being non empty set
for f being FinSequence of D
for d being Element of D
for i, j being Element of NAT st i <> j & j in dom f holds
(f +* (i,d)) /. j = f /. j
proof
let D be non empty set ; ::_thesis: for f being FinSequence of D
for d being Element of D
for i, j being Element of NAT st i <> j & j in dom f holds
(f +* (i,d)) /. j = f /. j
let f be FinSequence of D; ::_thesis: for d being Element of D
for i, j being Element of NAT st i <> j & j in dom f holds
(f +* (i,d)) /. j = f /. j
let d be Element of D; ::_thesis: for i, j being Element of NAT st i <> j & j in dom f holds
(f +* (i,d)) /. j = f /. j
let i, j be Element of NAT ; ::_thesis: ( i <> j & j in dom f implies (f +* (i,d)) /. j = f /. j )
assume that
A1: i <> j and
A2: j in dom f ; ::_thesis: (f +* (i,d)) /. j = f /. j
 j in dom (f +* (i,d)) by A2, Th30;
hence (f +* (i,d)) /. j = (f +* (i,d)) . j by PARTFUN1:def_6
.= f . j by A1, Th32
.= f /. j by A2, PARTFUN1:def_6 ;
::_thesis: verum
end;

theorem :: FUNCT_7:38
for D being non empty set
for f being FinSequence of D
for d, e being Element of D
for i being Element of NAT holds f +* (i,(f /. i)) = f
proof
let D be non empty set ; ::_thesis: for f being FinSequence of D
for d, e being Element of D
for i being Element of NAT holds f +* (i,(f /. i)) = f
let f be FinSequence of D; ::_thesis: for d, e being Element of D
for i being Element of NAT holds f +* (i,(f /. i)) = f
let d, e be Element of D; ::_thesis: for i being Element of NAT holds f +* (i,(f /. i)) = f
let i be Element of NAT ; ::_thesis: f +* (i,(f /. i)) = f
percases ( i in dom f or not i in dom f ) ;
suppose i in dom f ; ::_thesis: f +* (i,(f /. i)) = f
hence f +* (i,(f /. i)) = f +* (i,(f . i)) by PARTFUN1:def_6
.= f by Th35 ;
::_thesis: verum
end;
suppose not i in dom f ; ::_thesis: f +* (i,(f /. i)) = f
hence  f +* (i,(f /. i)) = f by Def3; ::_thesis: verum
end;
end;
end;

begin

definition
let X be set ;
let p be Function-yielding FinSequence;
func compose (p,X) ->  Function means :Def4: :: FUNCT_7:def 4
 ex f being ManySortedFunction of NAT st
( it = f . (len p) & f . 0 = id X & ( for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g ) );
uniqueness
 for b1, b2 being Function st ex f being ManySortedFunction of NAT st
( b1 = f . (len p) & f . 0 = id X & ( for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g ) ) & ex f being ManySortedFunction of NAT st
( b2 = f . (len p) & f . 0 = id X & ( for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g ) ) holds
b1 = b2
proof
let g1, g2 be Function; ::_thesis: ( ex f being ManySortedFunction of NAT st
( g1 = f . (len p) & f . 0 = id X & ( for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g ) ) & ex f being ManySortedFunction of NAT st
( g2 = f . (len p) & f . 0 = id X & ( for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g ) ) implies g1 = g2 )
given f1 being ManySortedFunction of NAT  such that A1: g1 = f1 . (len p) and
A2: f1 . 0 = id X and
A3: for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f1 . i & h = p . (i + 1) holds
f1 . (i + 1) = h * g ; ::_thesis: ( for f being ManySortedFunction of NAT holds
( not g2 = f . (len p) or not f . 0 = id X or ex i being Element of NAT st
( i + 1 in dom p & ex g, h being Function st
( g = f . i & h = p . (i + 1) & not f . (i + 1) = h * g ) ) ) or g1 = g2 )
given f2 being ManySortedFunction of NAT  such that A4: g2 = f2 . (len p) and
A5: f2 . 0 = id X and
A6: for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f2 . i & h = p . (i + 1) holds
f2 . (i + 1) = h * g ; ::_thesis: g1 = g2
defpred S1[ Element of NAT ] means ( $1 <= len p implies ( f1 . $1 = f2 . $1 & f1 . $1 is Function ) );
A7: for i being Element of NAT st S1[i] holds
S1[i + 1]
proof
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume that
A8: ( i <= len p implies ( f1 . i = f2 . i & f1 . i is Function ) ) and
A9: i + 1 <= len p ; ::_thesis: ( f1 . (i + 1) = f2 . (i + 1) & f1 . (i + 1) is Function )
reconsider h = p . (i + 1) as Function ;
reconsider g = f1 . i as Function ;
 i + 1 >= 1 by NAT_1:11;
then A10: i + 1 in dom p by A9, FINSEQ_3:25;
then  f1 . (i + 1) = h * g by A3;
hence  ( f1 . (i + 1) = f2 . (i + 1) & f1 . (i + 1) is Function ) by A6, A8, A9, A10, NAT_1:13; ::_thesis: verum
end;
A11: S1[ 0 ] by A2, A5;
 for i being Element of NAT holds S1[i] from NAT_1:sch_1(A11, A7);
hence  g1 = g2 by A1, A4; ::_thesis: verum
end;
correctness
existence
ex b1 being Function ex f being ManySortedFunction of NAT st
( b1 = f . (len p) & f . 0 = id X & ( for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g ) );
proof
reconsider e = 0 as Function ;
defpred S1[ Element of NAT , set , set ] means ( ( $2 is not Function & $3 = e ) or ( $2 is Function & ( for g, h being Function st g = $2 & h = p . ($1 + 1) holds
$3 = h * g ) ) );
A12: for n being Element of NAT
for x being set ex y being set st S1[n,x,y]
proof
let n be Element of NAT ; ::_thesis: for x being set ex y being set st S1[n,x,y]
let x be set ; ::_thesis: ex y being set st S1[n,x,y]
reconsider h = p . (n + 1) as Function ;
percases ( x is Function or not x is Function ) ;
suppose x is Function ; ::_thesis: ex y being set st S1[n,x,y]
then reconsider g = x as Function ;
take  h * g ; ::_thesis: S1[n,x,h * g]
thus  S1[n,x,h * g] ; ::_thesis: verum
end;
supposeA13: x is not Function ; ::_thesis: ex y being set st S1[n,x,y]
take  0 ; ::_thesis: S1[n,x, 0 ]
thus  S1[n,x, 0 ] by A13; ::_thesis: verum
end;
end;
end;
consider f being Function such that
A14: ( dom f = NAT & f . 0 = id X ) and
A15: for n being Element of NAT holds S1[n,f . n,f . (n + 1)] from RECDEF_1:sch_1(A12);
defpred S2[ Element of NAT ] means f . $1 is Function;
A16: for i being Element of NAT st S2[i] holds
S2[i + 1]
proof
let i be Element of NAT ; ::_thesis: ( S2[i] implies S2[i + 1] )
reconsider h = p . (i + 1) as Function ;
assume  f . i is Function ; ::_thesis: S2[i + 1]
then reconsider g = f . i as Function ;
 f . (i + 1) = h * g by A15;
hence  S2[i + 1] ; ::_thesis: verum
end;
A17: S2[ 0 ] by A14;
A18: for i being Element of NAT holds S2[i] from NAT_1:sch_1(A17, A16);
then reconsider F = f . (len p) as Function ;
 f is Function-yielding
proof
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 f or f . x is set )
assume  x in dom f ; ::_thesis: f . x is set
hence  f . x is set by A14, A18; ::_thesis: verum
end;
then reconsider f = f as ManySortedFunction of NAT by A14, PARTFUN1:def_2, RELAT_1:def_18;
take  F ; ::_thesis: ex f being ManySortedFunction of NAT st
( F = f . (len p) & f . 0 = id X & ( for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g ) )
take  f ; ::_thesis: ( F = f . (len p) & f . 0 = id X & ( for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g ) )
thus  ( F = f . (len p) & f . 0 = id X ) by A14; ::_thesis: for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g
let i be Element of NAT ; ::_thesis: ( i + 1 in dom p implies for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g )
thus  ( i + 1 in dom p implies for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g ) by A15; ::_thesis: verum
end;
end;

:: deftheorem Def4 defines compose FUNCT_7:def_4_:_
 for X being set
for p being Function-yielding FinSequence
for b3 being Function holds
( b3 = compose (p,X) iff ex f being ManySortedFunction of NAT st
( b3 = f . (len p) & f . 0 = id X & ( for i being Element of NAT st i + 1 in dom p holds
for g, h being Function st g = f . i & h = p . (i + 1) holds
f . (i + 1) = h * g ) ) );

definition
let p be Function-yielding FinSequence;
let x be set ;
func apply (p,x) ->  FinSequence means :Def5: :: FUNCT_7:def 5
( len it = (len p) + 1 & it . 1 = x & ( for i being Element of NAT
for f being Function st i in dom p & f = p . i holds
it . (i + 1) = f . (it . i) ) );
existence
 ex b1 being FinSequence st
( len b1 = (len p) + 1 & b1 . 1 = x & ( for i being Element of NAT
for f being Function st i in dom p & f = p . i holds
b1 . (i + 1) = f . (b1 . i) ) )
proof
defpred S1[ Element of NAT , set , set ] means  ex f being Function st
( f = p . $1 & $3 = f . $2 );
A1: for i being Element of NAT st 1 <= i & i < (len p) + 1 holds
for x being set ex y being set st S1[i,x,y]
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < (len p) + 1 implies for x being set ex y being set st S1[i,x,y] )
assume  1 <= i ; ::_thesis: ( not i < (len p) + 1 or for x being set ex y being set st S1[i,x,y] )
assume  i < (len p) + 1 ; ::_thesis: for x being set ex y being set st S1[i,x,y]
reconsider f = p . i as Function ;
let x be set ; ::_thesis: ex y being set st S1[i,x,y]
take  f . x ; ::_thesis: S1[i,x,f . x]
take  f ; ::_thesis: ( f = p . i & f . x = f . x )
thus  ( f = p . i & f . x = f . x ) ; ::_thesis: verum
end;
consider q being FinSequence such that
A2: ( len q = (len p) + 1 & ( q . 1 = x or (len p) + 1 = 0 ) ) and
A3: for i being Element of NAT st 1 <= i & i < (len p) + 1 holds
S1[i,q . i,q . (i + 1)] from RECDEF_1:sch_3(A1);
take  q ; ::_thesis: ( len q = (len p) + 1 & q . 1 = x & ( for i being Element of NAT
for f being Function st i in dom p & f = p . i holds
q . (i + 1) = f . (q . i) ) )
thus  ( len q = (len p) + 1 & q . 1 = x ) by A2; ::_thesis: for i being Element of NAT
for f being Function st i in dom p & f = p . i holds
q . (i + 1) = f . (q . i)
let i be Element of NAT ; ::_thesis: for f being Function st i in dom p & f = p . i holds
q . (i + 1) = f . (q . i)
let f be Function; ::_thesis: ( i in dom p & f = p . i implies q . (i + 1) = f . (q . i) )
assume A4: i in dom p ; ::_thesis: ( not f = p . i or q . (i + 1) = f . (q . i) )
then  i <= len p by FINSEQ_3:25;
then A5: i < (len p) + 1 by NAT_1:13;
 i >= 1 by A4, FINSEQ_3:25;
then  ex f being Function st
( f = p . i & q . (i + 1) = f . (q . i) ) by A3, A5;
hence  ( not f = p . i or q . (i + 1) = f . (q . i) ) ; ::_thesis: verum
end;
correctness
uniqueness
for b1, b2 being FinSequence st len b1 = (len p) + 1 & b1 . 1 = x & ( for i being Element of NAT
for f being Function st i in dom p & f = p . i holds
b1 . (i + 1) = f . (b1 . i) ) & len b2 = (len p) + 1 & b2 . 1 = x & ( for i being Element of NAT
for f being Function st i in dom p & f = p . i holds
b2 . (i + 1) = f . (b2 . i) ) holds
b1 = b2;
proof
let q1, q2 be FinSequence; ::_thesis: ( len q1 = (len p) + 1 & q1 . 1 = x & ( for i being Element of NAT
for f being Function st i in dom p & f = p . i holds
q1 . (i + 1) = f . (q1 . i) ) & len q2 = (len p) + 1 & q2 . 1 = x & ( for i being Element of NAT
for f being Function st i in dom p & f = p . i holds
q2 . (i + 1) = f . (q2 . i) ) implies q1 = q2 )
assume that
A6: len q1 = (len p) + 1 and
A7: q1 . 1 = x and
A8: for i being Element of NAT
for f being Function st i in dom p & f = p . i holds
q1 . (i + 1) = f . (q1 . i) and
A9: len q2 = (len p) + 1 and
A10: q2 . 1 = x and
A11: for i being Element of NAT
for f being Function st i in dom p & f = p . i holds
q2 . (i + 1) = f . (q2 . i) ; ::_thesis: q1 = q2
defpred S1[ Nat] means ( $1 in dom q1 implies q1 . $1 = q2 . $1 );
A12: for i being Nat st S1[i] holds
S1[i + 1]
proof
let i be Nat; ::_thesis: ( S1[i] implies S1[i + 1] )
assume that
A13: ( i in dom q1 implies q1 . i = q2 . i ) and
A14: i + 1 in dom q1 ; ::_thesis: q1 . (i + 1) = q2 . (i + 1)
 i + 1 <= len q1 by A14, FINSEQ_3:25;
then A15: i <= len p by A6, XREAL_1:6;
percases ( i = 0 or i > 0 ) ;
suppose i = 0 ; ::_thesis: q1 . (i + 1) = q2 . (i + 1)
hence  q1 . (i + 1) = q2 . (i + 1) by A7, A10; ::_thesis: verum
end;
supposeA16: i > 0 ; ::_thesis: q1 . (i + 1) = q2 . (i + 1)
reconsider f = p . i as Function ;
 i >= 0 + 1 by A16, NAT_1:13;
then A17: i in dom p by A15, FINSEQ_3:25;
hence q1 . (i + 1) = f . (q2 . i) by A6, A8, A13, Th22
.= q2 . (i + 1) by A11, A17 ;
::_thesis: verum
end;
end;
end;
A18: S1[ 0 ] by FINSEQ_3:25;
A19: for i being Nat holds S1[i] from NAT_1:sch_2(A18, A12);
 dom q1 = dom q2 by A6, A9, FINSEQ_3:29;
hence  q1 = q2 by A19, FINSEQ_1:13; ::_thesis: verum
end;
end;

:: deftheorem Def5 defines apply FUNCT_7:def_5_:_
 for p being Function-yielding FinSequence
for x being set
for b3 being FinSequence holds
( b3 = apply (p,x) iff ( len b3 = (len p) + 1 & b3 . 1 = x & ( for i being Element of NAT
for f being Function st i in dom p & f = p . i holds
b3 . (i + 1) = f . (b3 . i) ) ) );


theorem Th39: :: FUNCT_7:39
for X being set holds compose ({},X) = id X
proof
let X be set ; ::_thesis: compose ({},X) = id X
 ex f being ManySortedFunction of NAT st
( compose ({},X) = f . 0 & f . 0 = id X & ( for i being Element of NAT st i + 1 in dom {} holds
for g, h being Function st g = f . i & h = {} . (i + 1) holds
f . (i + 1) = h * g ) ) by Def4, CARD_1:27;
hence  compose ({},X) = id X ; ::_thesis: verum
end;

theorem Th40: :: FUNCT_7:40
for x being set holds apply ({},x) = <*x*>
proof
let x be set ; ::_thesis: apply ({},x) = <*x*>
 ( len (apply ({},x)) = 0 + 1 & (apply ({},x)) . 1 = x ) by Def5, CARD_1:27;
hence  apply ({},x) = <*x*> by FINSEQ_1:40; ::_thesis: verum
end;

theorem Th41: :: FUNCT_7:41
for X being set
for p being Function-yielding FinSequence
for f being Function holds compose ((p ^ <*f*>),X) = f * (compose (p,X))
proof
let X be set ; ::_thesis: for p being Function-yielding FinSequence
for f being Function holds compose ((p ^ <*f*>),X) = f * (compose (p,X))
let p be Function-yielding FinSequence; ::_thesis: for f being Function holds compose ((p ^ <*f*>),X) = f * (compose (p,X))
let f be Function; ::_thesis: compose ((p ^ <*f*>),X) = f * (compose (p,X))
A1: f = (p ^ <*f*>) . ((len p) + 1) by FINSEQ_1:42;
 len <*f*> = 1 by FINSEQ_1:40;
then A2: len (p ^ <*f*>) = (len p) + 1 by FINSEQ_1:22;
consider ff being ManySortedFunction of NAT  such that
A3: compose ((p ^ <*f*>),X) = ff . (len (p ^ <*f*>)) and
A4: ff . 0 = id X and
A5: for i being Element of NAT st i + 1 in dom (p ^ <*f*>) holds
for g, h being Function st g = ff . i & h = (p ^ <*f*>) . (i + 1) holds
ff . (i + 1) = h * g by Def4;
reconsider g = ff . (len p) as Function ;
A6: dom p c= dom (p ^ <*f*>) by FINSEQ_1:26;
now__::_thesis:_for_i_being_Element_of_NAT_st_i_+_1_in_dom_p_holds_
for_g,_h_being_Function_st_g_=_ff_._i_&_h_=_p_._(i_+_1)_holds_
ff_._(i_+_1)_=_h_*_g
let i be Element of NAT ; ::_thesis: ( i + 1 in dom p implies for g, h being Function st g = ff . i & h = p . (i + 1) holds
ff . (i + 1) = h * g )
assume A7: i + 1 in dom p ; ::_thesis: for g, h being Function st g = ff . i & h = p . (i + 1) holds
ff . (i + 1) = h * g
let g, h be Function; ::_thesis: ( g = ff . i & h = p . (i + 1) implies ff . (i + 1) = h * g )
assume that
A8: g = ff . i and
A9: h = p . (i + 1) ; ::_thesis: ff . (i + 1) = h * g
 h = (p ^ <*f*>) . (i + 1) by A7, A9, FINSEQ_1:def_7;
hence  ff . (i + 1) = h * g by A5, A6, A7, A8; ::_thesis: verum
end;
then A10: g = compose (p,X) by A4, Def4;
 ( 1 in Seg 1 & dom <*f*> = Seg 1 ) by FINSEQ_1:2, FINSEQ_1:38, TARSKI:def_1;
hence  compose ((p ^ <*f*>),X) = f * (compose (p,X)) by A3, A5, A10, A2, A1, FINSEQ_1:28; ::_thesis: verum
end;

theorem Th42: :: FUNCT_7:42
for x being set
for p being Function-yielding FinSequence
for f being Function holds apply ((p ^ <*f*>),x) = (apply (p,x)) ^ <*(f . ((apply (p,x)) . ((len p) + 1)))*>
proof
let x be set ; ::_thesis: for p being Function-yielding FinSequence
for f being Function holds apply ((p ^ <*f*>),x) = (apply (p,x)) ^ <*(f . ((apply (p,x)) . ((len p) + 1)))*>
let p be Function-yielding FinSequence; ::_thesis: for f being Function holds apply ((p ^ <*f*>),x) = (apply (p,x)) ^ <*(f . ((apply (p,x)) . ((len p) + 1)))*>
let f be Function; ::_thesis: apply ((p ^ <*f*>),x) = (apply (p,x)) ^ <*(f . ((apply (p,x)) . ((len p) + 1)))*>
defpred S1[ Nat] means ( $1 in dom (apply (p,x)) implies (apply ((p ^ <*f*>),x)) . $1 = (apply (p,x)) . $1 );
A1: len (apply (p,x)) = (len p) + 1 by Def5;
A2: (apply (p,x)) . 1 = x by Def5;
A3: for i being Nat st S1[i] holds
S1[i + 1]
proof
let i be Nat; ::_thesis: ( S1[i] implies S1[i + 1] )
assume that
A4: ( i in dom (apply (p,x)) implies (apply ((p ^ <*f*>),x)) . i = (apply (p,x)) . i ) and
A5: i + 1 in dom (apply (p,x)) ; ::_thesis: (apply ((p ^ <*f*>),x)) . (i + 1) = (apply (p,x)) . (i + 1)
A6: i + 1 <= len (apply (p,x)) by A5, FINSEQ_3:25;
then A7: i <= len (apply (p,x)) by NAT_1:13;
A8: i <= len p by A1, A6, XREAL_1:6;
percases ( i = 0 or i > 0 ) ;
suppose i = 0 ; ::_thesis: (apply ((p ^ <*f*>),x)) . (i + 1) = (apply (p,x)) . (i + 1)
hence  (apply ((p ^ <*f*>),x)) . (i + 1) = (apply (p,x)) . (i + 1) by A2, Def5; ::_thesis: verum
end;
supposeA9: i > 0 ; ::_thesis: (apply ((p ^ <*f*>),x)) . (i + 1) = (apply (p,x)) . (i + 1)
reconsider g = p . i as Function ;
A10: i >= 0 + 1 by A9, NAT_1:13;
then A11: i in dom p by A8, FINSEQ_3:25;
then  ( dom p c= dom (p ^ <*f*>) & g = (p ^ <*f*>) . i ) by FINSEQ_1:26, FINSEQ_1:def_7;
then  (apply ((p ^ <*f*>),x)) . (i + 1) = g . ((apply ((p ^ <*f*>),x)) . i) by A11, Def5;
hence  (apply ((p ^ <*f*>),x)) . (i + 1) = (apply (p,x)) . (i + 1) by A4, A7, A10, A11, Def5, FINSEQ_3:25; ::_thesis: verum
end;
end;
end;
A12: S1[ 0 ] by FINSEQ_3:25;
A13: for i being Nat holds S1[i] from NAT_1:sch_2(A12, A3);
 len <*f*> = 1 by FINSEQ_1:40;
then A14: len (p ^ <*f*>) = (len p) + 1 by FINSEQ_1:22;
A15: (len p) + 1 >= 1 by NAT_1:11;
then A16: ( (p ^ <*f*>) . ((len p) + 1) = f & (len p) + 1 in dom (p ^ <*f*>) ) by A14, FINSEQ_1:42, FINSEQ_3:25;
A17: (len p) + 1 in dom (apply (p,x)) by A1, A15, FINSEQ_3:25;
A18: now__::_thesis:_for_i_being_Nat_st_i_in_dom_<*(f_._((apply_(p,x))_._((len_p)_+_1)))*>_holds_
(apply_((p_^_<*f*>),x))_._((len_(apply_(p,x)))_+_i)_=_<*(f_._((apply_(p,x))_._((len_p)_+_1)))*>_._i
let i be Nat; ::_thesis: ( i in dom <*(f . ((apply (p,x)) . ((len p) + 1)))*> implies (apply ((p ^ <*f*>),x)) . ((len (apply (p,x))) + i) = <*(f . ((apply (p,x)) . ((len p) + 1)))*> . i )
assume  i in dom <*(f . ((apply (p,x)) . ((len p) + 1)))*> ; ::_thesis: (apply ((p ^ <*f*>),x)) . ((len (apply (p,x))) + i) = <*(f . ((apply (p,x)) . ((len p) + 1)))*> . i
then  i in Seg 1 by FINSEQ_1:38;
then A19: i = 1 by FINSEQ_1:2, TARSKI:def_1;
then  ( f . ((apply ((p ^ <*f*>),x)) . ((len p) + i)) = (apply ((p ^ <*f*>),x)) . ((len (apply (p,x))) + i) & (apply ((p ^ <*f*>),x)) . ((len p) + i) = (apply (p,x)) . ((len p) + i) ) by A1, A16, A17, A13, Def5;
hence  (apply ((p ^ <*f*>),x)) . ((len (apply (p,x))) + i) = <*(f . ((apply (p,x)) . ((len p) + 1)))*> . i by A19, FINSEQ_1:40; ::_thesis: verum
end;
 len (apply ((p ^ <*f*>),x)) = (len (p ^ <*f*>)) + 1 by Def5;
then  ( len <*(f . ((apply (p,x)) . ((len p) + 1)))*> = 1 & dom (apply ((p ^ <*f*>),x)) = Seg ((len (apply (p,x))) + 1) ) by A1, A14, FINSEQ_1:40, FINSEQ_1:def_3;
hence  apply ((p ^ <*f*>),x) = (apply (p,x)) ^ <*(f . ((apply (p,x)) . ((len p) + 1)))*> by A13, A18, FINSEQ_1:def_7; ::_thesis: verum
end;

theorem :: FUNCT_7:43
for X being set
for p being Function-yielding FinSequence
for f being Function holds compose ((<*f*> ^ p),X) = (compose (p,(f .: X))) * (f | X)
proof
let X be set ; ::_thesis: for p being Function-yielding FinSequence
for f being Function holds compose ((<*f*> ^ p),X) = (compose (p,(f .: X))) * (f | X)
let p be Function-yielding FinSequence; ::_thesis: for f being Function holds compose ((<*f*> ^ p),X) = (compose (p,(f .: X))) * (f | X)
let f be Function; ::_thesis: compose ((<*f*> ^ p),X) = (compose (p,(f .: X))) * (f | X)
defpred S1[ Function-yielding FinSequence] means  compose ((<*f*> ^ $1),X) = (compose ($1,(f .: X))) * (f | X);
A1: for p being Function-yielding FinSequence st S1[p] holds
for f being Function holds S1[p ^ <*f*>]
proof
let p be Function-yielding FinSequence; ::_thesis: ( S1[p] implies for f being Function holds S1[p ^ <*f*>] )
assume A2: compose ((<*f*> ^ p),X) = (compose (p,(f .: X))) * (f | X) ; ::_thesis: for f being Function holds S1[p ^ <*f*>]
let g be Function; ::_thesis: S1[p ^ <*g*>]
thus  compose ((<*f*> ^ (p ^ <*g*>)),X) = compose (((<*f*> ^ p) ^ <*g*>),X) by FINSEQ_1:32
.= g * (compose ((<*f*> ^ p),X)) by Th41
.= (g * (compose (p,(f .: X)))) * (f | X) by A2, RELAT_1:36
.= (compose ((p ^ <*g*>),(f .: X))) * (f | X) by Th41 ; ::_thesis: verum
end;
 ( <*f*> ^ {} = <*f*> & {} ^ <*f*> = <*f*> ) by FINSEQ_1:34;
then  compose ((<*f*> ^ {}),X) = f * (compose ({},X)) by Th41
.= f * (id X) by Th39
.= f | X by RELAT_1:65
.= (id (rng (f | X))) * (f | X) by RELAT_1:54
.= (id (f .: X)) * (f | X) by RELAT_1:115
.= (compose ({},(f .: X))) * (f | X) by Th39 ;
then A3: S1[ {} ] ;
 for p being Function-yielding FinSequence holds S1[p] from FUNCT_7:sch_5(A3, A1);
hence  compose ((<*f*> ^ p),X) = (compose (p,(f .: X))) * (f | X) ; ::_thesis: verum
end;

theorem :: FUNCT_7:44
for x being set
for p being Function-yielding FinSequence
for f being Function holds apply ((<*f*> ^ p),x) = <*x*> ^ (apply (p,(f . x)))
proof
let x be set ; ::_thesis: for p being Function-yielding FinSequence
for f being Function holds apply ((<*f*> ^ p),x) = <*x*> ^ (apply (p,(f . x)))
let p be Function-yielding FinSequence; ::_thesis: for f being Function holds apply ((<*f*> ^ p),x) = <*x*> ^ (apply (p,(f . x)))
let f be Function; ::_thesis: apply ((<*f*> ^ p),x) = <*x*> ^ (apply (p,(f . x)))
defpred S1[ Function-yielding FinSequence] means  apply ((<*f*> ^ $1),x) = <*x*> ^ (apply ($1,(f . x)));
A1: len {} = 0 ;
A2: for p being Function-yielding FinSequence st S1[p] holds
for f being Function holds S1[p ^ <*f*>]
proof
let p be Function-yielding FinSequence; ::_thesis: ( S1[p] implies for f being Function holds S1[p ^ <*f*>] )
assume A3: apply ((<*f*> ^ p),x) = <*x*> ^ (apply (p,(f . x))) ; ::_thesis: for f being Function holds S1[p ^ <*f*>]
let g be Function; ::_thesis: S1[p ^ <*g*>]
set p9 = <*f*> ^ p;
A4: len (apply (p,(f . x))) = (len p) + 1 by Def5;
 len <*f*> = 1 by FINSEQ_1:40;
then A5: len (<*f*> ^ p) = (len p) + 1 by FINSEQ_1:22;
then  len (<*f*> ^ p) >= 1 by NAT_1:11;
then  ( len <*x*> = 1 & len (<*f*> ^ p) in dom (apply (p,(f . x))) ) by A4, A5, FINSEQ_1:40, FINSEQ_3:25;
then A6: (apply ((<*f*> ^ p),x)) . (1 + (len (<*f*> ^ p))) = (apply (p,(f . x))) . ((len p) + 1) by A3, A5, FINSEQ_1:def_7;
 apply (((<*f*> ^ p) ^ <*g*>),x) = (apply ((<*f*> ^ p),x)) ^ <*(g . ((apply ((<*f*> ^ p),x)) . ((len (<*f*> ^ p)) + 1)))*> by Th42;
hence  apply ((<*f*> ^ (p ^ <*g*>)),x) = (<*x*> ^ (apply (p,(f . x)))) ^ <*(g . ((apply (p,(f . x))) . ((len p) + 1)))*> by A3, A6, FINSEQ_1:32
.= <*x*> ^ ((apply (p,(f . x))) ^ <*(g . ((apply (p,(f . x))) . ((len p) + 1)))*>) by FINSEQ_1:32
.= <*x*> ^ (apply ((p ^ <*g*>),(f . x))) by Th42 ;
::_thesis: verum
end;
 ( <*f*> ^ {} = <*f*> & {} ^ <*f*> = <*f*> ) by FINSEQ_1:34;
then  apply ((<*f*> ^ {}),x) = (apply ({},x)) ^ <*(f . ((apply ({},x)) . (0 + 1)))*> by A1, Th42
.= <*x*> ^ <*(f . ((apply ({},x)) . 1))*> by Th40
.= <*x*> ^ <*(f . (<*x*> . 1))*> by Th40
.= <*x*> ^ <*(f . x)*> by FINSEQ_1:40
.= <*x*> ^ (apply ({},(f . x))) by Th40 ;
then A7: S1[ {} ] ;
 for p being Function-yielding FinSequence holds S1[p] from FUNCT_7:sch_5(A7, A2);
hence  apply ((<*f*> ^ p),x) = <*x*> ^ (apply (p,(f . x))) ; ::_thesis: verum
end;

theorem Th45: :: FUNCT_7:45
for X being set
for f being Function holds compose (<*f*>,X) = f * (id X)
proof
let X be set ; ::_thesis: for f being Function holds compose (<*f*>,X) = f * (id X)
let f be Function; ::_thesis: compose (<*f*>,X) = f * (id X)
 <*f*> = {} ^ <*f*> by FINSEQ_1:34;
hence  compose (<*f*>,X) = f * (compose ({},X)) by Th41
.= f * (id X) by Th39 ;
::_thesis: verum
end;

theorem :: FUNCT_7:46
for X being set
for f being Function st dom f c= X holds
compose (<*f*>,X) = f
proof
let X be set ; ::_thesis: for f being Function st dom f c= X holds
compose (<*f*>,X) = f
let f be Function; ::_thesis: ( dom f c= X implies compose (<*f*>,X) = f )
 compose (<*f*>,X) = f * (id X) by Th45;
hence  ( dom f c= X implies compose (<*f*>,X) = f ) by RELAT_1:51; ::_thesis: verum
end;

theorem Th47: :: FUNCT_7:47
for x being set
for f being Function holds apply (<*f*>,x) = <*x,(f . x)*>
proof
let x be set ; ::_thesis: for f being Function holds apply (<*f*>,x) = <*x,(f . x)*>
let f be Function; ::_thesis: apply (<*f*>,x) = <*x,(f . x)*>
A1: <*x*> . (0 + 1) = x by FINSEQ_1:40;
A2: ( apply ({},x) = <*x*> & len {} = 0 ) by Th40;
thus  apply (<*f*>,x) = apply (({} ^ <*f*>),x) by FINSEQ_1:34
.= <*x*> ^ <*(f . x)*> by A2, A1, Th42
.= <*x,(f . x)*> by FINSEQ_1:def_9 ; ::_thesis: verum
end;

theorem :: FUNCT_7:48
for X, Y being set
for p, q being Function-yielding FinSequence st rng (compose (p,X)) c= Y holds
compose ((p ^ q),X) = (compose (q,Y)) * (compose (p,X))
proof
let X, Y be set ; ::_thesis: for p, q being Function-yielding FinSequence st rng (compose (p,X)) c= Y holds
compose ((p ^ q),X) = (compose (q,Y)) * (compose (p,X))
let p, q be Function-yielding FinSequence; ::_thesis: ( rng (compose (p,X)) c= Y implies compose ((p ^ q),X) = (compose (q,Y)) * (compose (p,X)) )
assume A1: rng (compose (p,X)) c= Y ; ::_thesis: compose ((p ^ q),X) = (compose (q,Y)) * (compose (p,X))
defpred S1[ Function-yielding FinSequence] means  compose ((p ^ $1),X) = (compose ($1,Y)) * (compose (p,X));
A2: for p being Function-yielding FinSequence st S1[p] holds
for f being Function holds S1[p ^ <*f*>]
proof
let q be Function-yielding FinSequence; ::_thesis: ( S1[q] implies for f being Function holds S1[q ^ <*f*>] )
assume A3: compose ((p ^ q),X) = (compose (q,Y)) * (compose (p,X)) ; ::_thesis: for f being Function holds S1[q ^ <*f*>]
let f be Function; ::_thesis: S1[q ^ <*f*>]
thus  compose ((p ^ (q ^ <*f*>)),X) = compose (((p ^ q) ^ <*f*>),X) by FINSEQ_1:32
.= f * (compose ((p ^ q),X)) by Th41
.= (f * (compose (q,Y))) * (compose (p,X)) by A3, RELAT_1:36
.= (compose ((q ^ <*f*>),Y)) * (compose (p,X)) by Th41 ; ::_thesis: verum
end;
compose ((p ^ {}),X) = compose (p,X) by FINSEQ_1:34
.= (id Y) * (compose (p,X)) by A1, RELAT_1:53
.= (compose ({},Y)) * (compose (p,X)) by Th39 ;
then A4: S1[ {} ] ;
 for q being Function-yielding FinSequence holds S1[q] from FUNCT_7:sch_5(A4, A2);
hence  compose ((p ^ q),X) = (compose (q,Y)) * (compose (p,X)) ; ::_thesis: verum
end;

theorem Th49: :: FUNCT_7:49
for x being set
for p, q being Function-yielding FinSequence holds (apply ((p ^ q),x)) . ((len (p ^ q)) + 1) = (apply (q,((apply (p,x)) . ((len p) + 1)))) . ((len q) + 1)
proof
let x be set ; ::_thesis: for p, q being Function-yielding FinSequence holds (apply ((p ^ q),x)) . ((len (p ^ q)) + 1) = (apply (q,((apply (p,x)) . ((len p) + 1)))) . ((len q) + 1)
let p, q be Function-yielding FinSequence; ::_thesis: (apply ((p ^ q),x)) . ((len (p ^ q)) + 1) = (apply (q,((apply (p,x)) . ((len p) + 1)))) . ((len q) + 1)
defpred S1[ Function-yielding FinSequence] means (apply ((p ^ $1),x)) . ((len (p ^ $1)) + 1) = (apply ($1,((apply (p,x)) . ((len p) + 1)))) . ((len $1) + 1);
A1: for p being Function-yielding FinSequence st S1[p] holds
for f being Function holds S1[p ^ <*f*>]
proof
set y = (apply (p,x)) . ((len p) + 1);
let q be Function-yielding FinSequence; ::_thesis: ( S1[q] implies for f being Function holds S1[q ^ <*f*>] )
assume A2: (apply ((p ^ q),x)) . ((len (p ^ q)) + 1) = (apply (q,((apply (p,x)) . ((len p) + 1)))) . ((len q) + 1) ; ::_thesis: for f being Function holds S1[q ^ <*f*>]
let f be Function; ::_thesis: S1[q ^ <*f*>]
A3: len <*f*> = 1 by FINSEQ_1:40;
then A4: len ((p ^ q) ^ <*f*>) = (len (p ^ q)) + 1 by FINSEQ_1:22;
A5: len (q ^ <*f*>) = (len q) + 1 by A3, FINSEQ_1:22;
A6: ( apply ((q ^ <*f*>),((apply (p,x)) . ((len p) + 1))) = (apply (q,((apply (p,x)) . ((len p) + 1)))) ^ <*(f . ((apply (q,((apply (p,x)) . ((len p) + 1)))) . ((len q) + 1)))*> & len (apply (q,((apply (p,x)) . ((len p) + 1)))) = (len q) + 1 ) by Def5, Th42;
A7: len (apply ((p ^ q),x)) = (len (p ^ q)) + 1 by Def5;
 ( p ^ (q ^ <*f*>) = (p ^ q) ^ <*f*> & apply (((p ^ q) ^ <*f*>),x) = (apply ((p ^ q),x)) ^ <*(f . ((apply ((p ^ q),x)) . ((len (p ^ q)) + 1)))*> ) by Th42, FINSEQ_1:32;
hence (apply ((p ^ (q ^ <*f*>)),x)) . ((len (p ^ (q ^ <*f*>))) + 1) = f . ((apply ((p ^ q),x)) . ((len (p ^ q)) + 1)) by A7, A4, FINSEQ_1:42
.= (apply ((q ^ <*f*>),((apply (p,x)) . ((len p) + 1)))) . ((len (q ^ <*f*>)) + 1) by A2, A6, A5, FINSEQ_1:42 ;
::_thesis: verum
end;
 ( apply ({},((apply (p,x)) . ((len p) + 1))) = <*((apply (p,x)) . ((len p) + 1))*> & p ^ {} = p ) by Th40, FINSEQ_1:34;
then A8: S1[ {} ] by FINSEQ_1:40;
 for q being Function-yielding FinSequence holds S1[q] from FUNCT_7:sch_5(A8, A1);
hence  (apply ((p ^ q),x)) . ((len (p ^ q)) + 1) = (apply (q,((apply (p,x)) . ((len p) + 1)))) . ((len q) + 1) ; ::_thesis: verum
end;

theorem :: FUNCT_7:50
for x being set
for p, q being Function-yielding FinSequence holds apply ((p ^ q),x) = (apply (p,x)) $^ (apply (q,((apply (p,x)) . ((len p) + 1))))
proof
let x be set ; ::_thesis: for p, q being Function-yielding FinSequence holds apply ((p ^ q),x) = (apply (p,x)) $^ (apply (q,((apply (p,x)) . ((len p) + 1))))
let p, q be Function-yielding FinSequence; ::_thesis: apply ((p ^ q),x) = (apply (p,x)) $^ (apply (q,((apply (p,x)) . ((len p) + 1))))
defpred S1[ Function-yielding FinSequence] means  apply ((p ^ $1),x) = (apply (p,x)) $^ (apply ($1,((apply (p,x)) . ((len p) + 1))));
A1: len (apply (p,x)) = (len p) + 1 by Def5;
then consider r being FinSequence, y being set  such that
A2: apply (p,x) = r ^ <*y*> by CARD_1:27, FINSEQ_1:46;
A3: for p being Function-yielding FinSequence st S1[p] holds
for f being Function holds S1[p ^ <*f*>]
proof
let q be Function-yielding FinSequence; ::_thesis: ( S1[q] implies for f being Function holds S1[q ^ <*f*>] )
assume A4: apply ((p ^ q),x) = (apply (p,x)) $^ (apply (q,((apply (p,x)) . ((len p) + 1)))) ; ::_thesis: for f being Function holds S1[q ^ <*f*>]
A5: (apply ((p ^ q),x)) . ((len (p ^ q)) + 1) = (apply (q,((apply (p,x)) . ((len p) + 1)))) . ((len q) + 1) by Th49;
let f be Function; ::_thesis: S1[q ^ <*f*>]
A6: len (apply (q,((apply (p,x)) . ((len p) + 1)))) = (len q) + 1 by Def5;
A7: len (apply ((q ^ <*f*>),((apply (p,x)) . ((len p) + 1)))) = (len (q ^ <*f*>)) + 1 by Def5;
thus  apply ((p ^ (q ^ <*f*>)),x) = apply (((p ^ q) ^ <*f*>),x) by FINSEQ_1:32
.= (apply ((p ^ q),x)) ^ <*(f . ((apply ((p ^ q),x)) . ((len (p ^ q)) + 1)))*> by Th42
.= (r ^ (apply (q,((apply (p,x)) . ((len p) + 1))))) ^ <*(f . ((apply ((p ^ q),x)) . ((len (p ^ q)) + 1)))*> by A2, A4, A6, CARD_1:27, REWRITE1:2
.= r ^ ((apply (q,((apply (p,x)) . ((len p) + 1)))) ^ <*(f . ((apply ((p ^ q),x)) . ((len (p ^ q)) + 1)))*>) by FINSEQ_1:32
.= r ^ (apply ((q ^ <*f*>),((apply (p,x)) . ((len p) + 1)))) by A5, Th42
.= (apply (p,x)) $^ (apply ((q ^ <*f*>),((apply (p,x)) . ((len p) + 1)))) by A2, A7, CARD_1:27, REWRITE1:2 ; ::_thesis: verum
end;
 len <*y*> = 1 by FINSEQ_1:40;
then  (len p) + 1 = (len r) + 1 by A1, A2, FINSEQ_1:22;
then A8: (apply (p,x)) . ((len p) + 1) = y by A2, FINSEQ_1:42;
apply ((p ^ {}),x) = apply (p,x) by FINSEQ_1:34
.= (apply (p,x)) $^ <*((apply (p,x)) . ((len p) + 1))*> by A2, A8, REWRITE1:2
.= (apply (p,x)) $^ (apply ({},((apply (p,x)) . ((len p) + 1)))) by Th40 ;
then A9: S1[ {} ] ;
 for q being Function-yielding FinSequence holds S1[q] from FUNCT_7:sch_5(A9, A3);
hence  apply ((p ^ q),x) = (apply (p,x)) $^ (apply (q,((apply (p,x)) . ((len p) + 1)))) ; ::_thesis: verum
end;

theorem Th51: :: FUNCT_7:51
for X being set
for f, g being Function holds compose (<*f,g*>,X) = (g * f) * (id X)
proof
let X be set ; ::_thesis: for f, g being Function holds compose (<*f,g*>,X) = (g * f) * (id X)
let f, g be Function; ::_thesis: compose (<*f,g*>,X) = (g * f) * (id X)
 <*f,g*> = <*f*> ^ <*g*> by FINSEQ_1:def_9;
hence  compose (<*f,g*>,X) = g * (compose (<*f*>,X)) by Th41
.= g * (f * (id X)) by Th45
.= (g * f) * (id X) by RELAT_1:36 ;
::_thesis: verum
end;

theorem :: FUNCT_7:52
for X being set
for f, g being Function st ( dom f c= X or dom (g * f) c= X ) holds
compose (<*f,g*>,X) = g * f
proof
let X be set ; ::_thesis: for f, g being Function st ( dom f c= X or dom (g * f) c= X ) holds
compose (<*f,g*>,X) = g * f
let f, g be Function; ::_thesis: ( ( dom f c= X or dom (g * f) c= X ) implies compose (<*f,g*>,X) = g * f )
 ( compose (<*f,g*>,X) = (g * f) * (id X) & (g * f) * (id X) = g * (f * (id X)) ) by Th51, RELAT_1:36;
hence  ( ( dom f c= X or dom (g * f) c= X ) implies compose (<*f,g*>,X) = g * f ) by RELAT_1:51; ::_thesis: verum
end;

theorem Th53: :: FUNCT_7:53
for x being set
for f, g being Function holds apply (<*f,g*>,x) = <*x,(f . x),(g . (f . x))*>
proof
let x be set ; ::_thesis: for f, g being Function holds apply (<*f,g*>,x) = <*x,(f . x),(g . (f . x))*>
let f, g be Function; ::_thesis: apply (<*f,g*>,x) = <*x,(f . x),(g . (f . x))*>
A1: ( apply (<*f*>,x) = <*x,(f . x)*> & len <*f*> = 1 ) by Th47, FINSEQ_1:40;
thus  apply (<*f,g*>,x) = apply ((<*f*> ^ <*g*>),x) by FINSEQ_1:def_9
.= <*x,(f . x)*> ^ <*(g . (<*x,(f . x)*> . (1 + 1)))*> by A1, Th42
.= <*x,(f . x)*> ^ <*(g . (f . x))*> by FINSEQ_1:44
.= <*x,(f . x),(g . (f . x))*> by FINSEQ_1:43 ; ::_thesis: verum
end;

theorem Th54: :: FUNCT_7:54
for X being set
for f, g, h being Function holds compose (<*f,g,h*>,X) = ((h * g) * f) * (id X)
proof
let X be set ; ::_thesis: for f, g, h being Function holds compose (<*f,g,h*>,X) = ((h * g) * f) * (id X)
let f, g, h be Function; ::_thesis: compose (<*f,g,h*>,X) = ((h * g) * f) * (id X)
 <*f,g,h*> = <*f,g*> ^ <*h*> by FINSEQ_1:43;
hence  compose (<*f,g,h*>,X) = h * (compose (<*f,g*>,X)) by Th41
.= h * ((g * f) * (id X)) by Th51
.= (h * (g * f)) * (id X) by RELAT_1:36
.= ((h * g) * f) * (id X) by RELAT_1:36 ;
::_thesis: verum
end;

theorem :: FUNCT_7:55
for X being set
for f, g, h being Function st ( dom f c= X or dom (g * f) c= X or dom ((h * g) * f) c= X ) holds
compose (<*f,g,h*>,X) = (h * g) * f
proof
let X be set ; ::_thesis: for f, g, h being Function st ( dom f c= X or dom (g * f) c= X or dom ((h * g) * f) c= X ) holds
compose (<*f,g,h*>,X) = (h * g) * f
let f, g, h be Function; ::_thesis: ( ( dom f c= X or dom (g * f) c= X or dom ((h * g) * f) c= X ) implies compose (<*f,g,h*>,X) = (h * g) * f )
A1: ( (h * g) * (f * (id X)) = h * (g * (f * (id X))) & g * (f * (id X)) = (g * f) * (id X) ) by RELAT_1:36;
A2: h * (g * f) = (h * g) * f by RELAT_1:36;
 ( compose (<*f,g,h*>,X) = ((h * g) * f) * (id X) & ((h * g) * f) * (id X) = (h * g) * (f * (id X)) ) by Th54, RELAT_1:36;
hence  ( ( dom f c= X or dom (g * f) c= X or dom ((h * g) * f) c= X ) implies compose (<*f,g,h*>,X) = (h * g) * f ) by A1, A2, RELAT_1:51; ::_thesis: verum
end;

theorem :: FUNCT_7:56
for x being set
for f, g, h being Function holds apply (<*f,g,h*>,x) = <*x*> ^ <*(f . x),(g . (f . x)),(h . (g . (f . x)))*>
proof
let x be set ; ::_thesis: for f, g, h being Function holds apply (<*f,g,h*>,x) = <*x*> ^ <*(f . x),(g . (f . x)),(h . (g . (f . x)))*>
let f, g, h be Function; ::_thesis: apply (<*f,g,h*>,x) = <*x*> ^ <*(f . x),(g . (f . x)),(h . (g . (f . x)))*>
A1: ( apply (<*f,g*>,x) = <*x,(f . x),(g . (f . x))*> & len <*f,g*> = 2 ) by Th53, FINSEQ_1:44;
thus  apply (<*f,g,h*>,x) = apply ((<*f,g*> ^ <*h*>),x) by FINSEQ_1:43
.= <*x,(f . x),(g . (f . x))*> ^ <*(h . (<*x,(f . x),(g . (f . x))*> . (2 + 1)))*> by A1, Th42
.= <*x,(f . x),(g . (f . x))*> ^ <*(h . (g . (f . x)))*> by FINSEQ_1:45
.= (<*x*> ^ <*(f . x),(g . (f . x))*>) ^ <*(h . (g . (f . x)))*> by FINSEQ_1:43
.= <*x*> ^ (<*(f . x),(g . (f . x))*> ^ <*(h . (g . (f . x)))*>) by FINSEQ_1:32
.= <*x*> ^ <*(f . x),(g . (f . x)),(h . (g . (f . x)))*> by FINSEQ_1:43 ; ::_thesis: verum
end;

definition
let F be FinSequence;
func firstdom F ->  set  means :Def6: :: FUNCT_7:def 6
it is empty  if F is empty
otherwise it = proj1 (F . 1);
correctness
consistency
for b1 being set holds verum;
existence
( ( for b1 being set holds verum ) & ( F is empty implies ex b1 being set st b1 is empty ) & ( not F is empty implies ex b1 being set st b1 = proj1 (F . 1) ) );
uniqueness
for b1, b2 being set holds
( ( F is empty & b1 is empty & b2 is empty implies b1 = b2 ) & ( not F is empty & b1 = proj1 (F . 1) & b2 = proj1 (F . 1) implies b1 = b2 ) );
;
func lastrng F ->  set  means :Def7: :: FUNCT_7:def 7
it is empty  if F is empty
otherwise it = proj2 (F . (len F));
correctness
consistency
for b1 being set holds verum;
existence
( ( for b1 being set holds verum ) & ( F is empty implies ex b1 being set st b1 is empty ) & ( not F is empty implies ex b1 being set st b1 = proj2 (F . (len F)) ) );
uniqueness
for b1, b2 being set holds
( ( F is empty & b1 is empty & b2 is empty implies b1 = b2 ) & ( not F is empty & b1 = proj2 (F . (len F)) & b2 = proj2 (F . (len F)) implies b1 = b2 ) );
;
end;

:: deftheorem Def6 defines firstdom FUNCT_7:def_6_:_
 for F being FinSequence
for b2 being set holds
( ( F is empty implies ( b2 = firstdom F iff b2 is empty ) ) & ( not F is empty implies ( b2 = firstdom F iff b2 = proj1 (F . 1) ) ) );

:: deftheorem Def7 defines lastrng FUNCT_7:def_7_:_
 for F being FinSequence
for b2 being set holds
( ( F is empty implies ( b2 = lastrng F iff b2 is empty ) ) & ( not F is empty implies ( b2 = lastrng F iff b2 = proj2 (F . (len F)) ) ) );

theorem Th57: :: FUNCT_7:57
( firstdom {} = {} & lastrng {} = {} ) by Def6, Def7;

theorem Th58: :: FUNCT_7:58
for f being Function
for p being FinSequence holds
( firstdom (<*f*> ^ p) = dom f & lastrng (p ^ <*f*>) = rng f )
proof
let f be Function; ::_thesis: for p being FinSequence holds
( firstdom (<*f*> ^ p) = dom f & lastrng (p ^ <*f*>) = rng f )
let p be FinSequence; ::_thesis: ( firstdom (<*f*> ^ p) = dom f & lastrng (p ^ <*f*>) = rng f )
thus  firstdom (<*f*> ^ p) = proj1 ((<*f*> ^ p) . 1) by Def6
.= dom f by FINSEQ_1:41 ; ::_thesis: lastrng (p ^ <*f*>) = rng f
 len <*f*> = 1 by FINSEQ_1:40;
then  len (p ^ <*f*>) = (len p) + 1 by FINSEQ_1:22;
hence  lastrng (p ^ <*f*>) = proj2 ((p ^ <*f*>) . ((len p) + 1)) by Def7
.= rng f by FINSEQ_1:42 ;
::_thesis: verum
end;

theorem Th59: :: FUNCT_7:59
for X being set
for p being Function-yielding FinSequence st p <> {} holds
rng (compose (p,X)) c= lastrng p
proof
let X be set ; ::_thesis: for p being Function-yielding FinSequence st p <> {} holds
rng (compose (p,X)) c= lastrng p
defpred S1[ Function-yielding FinSequence] means ( $1 <> {} implies rng (compose ($1,X)) c= lastrng $1 );
A1: for p being Function-yielding FinSequence st S1[p] holds
for f being Function holds S1[p ^ <*f*>]
proof
let q be Function-yielding FinSequence; ::_thesis: ( S1[q] implies for f being Function holds S1[q ^ <*f*>] )
assume  ( q <> {} implies rng (compose (q,X)) c= lastrng q ) ; ::_thesis: for f being Function holds S1[q ^ <*f*>]
let f be Function; ::_thesis: S1[q ^ <*f*>]
assume  q ^ <*f*> <> {} ; ::_thesis: rng (compose ((q ^ <*f*>),X)) c= lastrng (q ^ <*f*>)
 compose ((q ^ <*f*>),X) = f * (compose (q,X)) by Th41;
then  rng (compose ((q ^ <*f*>),X)) c= rng f by RELAT_1:26;
hence  rng (compose ((q ^ <*f*>),X)) c= lastrng (q ^ <*f*>) by Th58; ::_thesis: verum
end;
A2: S1[ {} ] ;
thus  for p being Function-yielding FinSequence holds S1[p] from FUNCT_7:sch_5(A2, A1); ::_thesis: verum
end;

definition
let IT be FinSequence;
attrIT is FuncSeq-like  means :Def8: :: FUNCT_7:def 8
 ex p being FinSequence st
( len p = (len IT) + 1 & ( for i being Element of NAT st i in dom IT holds
IT . i in Funcs ((p . i),(p . (i + 1))) ) );
end;

:: deftheorem Def8 defines FuncSeq-like FUNCT_7:def_8_:_
 for IT being FinSequence holds
( IT is FuncSeq-like iff ex p being FinSequence st
( len p = (len IT) + 1 & ( for i being Element of NAT st i in dom IT holds
IT . i in Funcs ((p . i),(p . (i + 1))) ) ) );

theorem Th60: :: FUNCT_7:60
for p, q being FinSequence st p ^ q is FuncSeq-like holds
( p is FuncSeq-like & q is FuncSeq-like )
proof
let p, q be FinSequence; ::_thesis: ( p ^ q is FuncSeq-like implies ( p is FuncSeq-like & q is FuncSeq-like ) )
given pq being FinSequence such that A1: len pq = (len (p ^ q)) + 1 and
A2: for i being Element of NAT st i in dom (p ^ q) holds
(p ^ q) . i in Funcs ((pq . i),(pq . (i + 1))) ; :: according to FUNCT_7:def_8 ::_thesis: ( p is FuncSeq-like & q is FuncSeq-like )
reconsider p1 = pq | (Seg ((len p) + 1)), p2 = pq | (Seg (len p)) as FinSequence by FINSEQ_1:15;
A3: len (p ^ q) = (len p) + (len q) by FINSEQ_1:22;
then A4: len p <= len (p ^ q) by NAT_1:11;
 len (p ^ q) <= len pq by A1, NAT_1:11;
then  len p <= len pq by A4, XXREAL_0:2;
then A5: len p2 = len p by FINSEQ_1:17;
hereby  :: according to FUNCT_7:def_8 ::_thesis: q is FuncSeq-like
take p1 = p1; ::_thesis: ( len p1 = (len p) + 1 & ( for i being Element of NAT st i in dom p holds
p . i in Funcs ((p1 . i),(p1 . (i + 1))) ) )
 len p <= len (p ^ q) by A3, NAT_1:11;
then  (len p) + 1 <= len pq by A1, XREAL_1:6;
hence A6: len p1 = (len p) + 1 by FINSEQ_1:17; ::_thesis: for i being Element of NAT st i in dom p holds
p . i in Funcs ((p1 . i),(p1 . (i + 1)))
let i be Element of NAT ; ::_thesis: ( i in dom p implies p . i in Funcs ((p1 . i),(p1 . (i + 1))) )
assume A7: i in dom p ; ::_thesis: p . i in Funcs ((p1 . i),(p1 . (i + 1)))
then  (p ^ q) . i = p . i by FINSEQ_1:def_7;
then A8: p . i in Funcs ((pq . i),(pq . (i + 1))) by A2, A7, FINSEQ_3:22;
 i in dom p1 by A6, A7, Th22;
then A9: pq . i = p1 . i by FUNCT_1:47;
 i + 1 in dom p1 by A6, A7, Th22;
hence  p . i in Funcs ((p1 . i),(p1 . (i + 1))) by A8, A9, FUNCT_1:47; ::_thesis: verum
end;
consider q2 being FinSequence such that
A10: pq = p2 ^ q2 by FINSEQ_1:80;
take  q2 ; :: according to FUNCT_7:def_8 ::_thesis: ( len q2 = (len q) + 1 & ( for i being Element of NAT st i in dom q holds
q . i in Funcs ((q2 . i),(q2 . (i + 1))) ) )
 len pq = (len p2) + (len q2) by A10, FINSEQ_1:22;
hence  len q2 = (len q) + 1 by A1, A3, A5; ::_thesis: for i being Element of NAT st i in dom q holds
q . i in Funcs ((q2 . i),(q2 . (i + 1)))
let x be Element of NAT ; ::_thesis: ( x in dom q implies q . x in Funcs ((q2 . x),(q2 . (x + 1))) )
assume A11: x in dom q ; ::_thesis: q . x in Funcs ((q2 . x),(q2 . (x + 1)))
then  (p ^ q) . ((len p) + x) = q . x by FINSEQ_1:def_7;
then A12: q . x in Funcs ((pq . ((len p) + x)),(pq . (((len p) + x) + 1))) by A2, A11, FINSEQ_1:28;
A13: (len p) + ((len q) + 1) = (len p) + (len q2) by A1, A3, A10, A5, FINSEQ_1:22;
then  x + 1 in dom q2 by A11, Th22;
then A14: q2 . (x + 1) = pq . ((len p) + (x + 1)) by A10, A5, FINSEQ_1:def_7;
 x in dom q2 by A13, A11, Th22;
hence  q . x in Funcs ((q2 . x),(q2 . (x + 1))) by A10, A5, A12, A14, FINSEQ_1:def_7; ::_thesis: verum
end;

registration
cluster Relation-like Function-like FinSequence-like FuncSeq-like  ->  Function-yielding for set ;
coherence
 for b1 being FinSequence st b1 is FuncSeq-like holds
b1 is Function-yielding
proof
let q be FinSequence; ::_thesis: ( q is FuncSeq-like implies q is Function-yielding )
given p being FinSequence such that  len p = (len q) + 1 and
A1: for i being Element of NAT st i in dom q holds
q . i in Funcs ((p . i),(p . (i + 1))) ; :: according to FUNCT_7:def_8 ::_thesis: q is Function-yielding
let i be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not i in proj1 q or q . i is set )
assume A2: i in dom q ; ::_thesis: q . i is set
then reconsider i = i as Element of NAT ;
 q . i in Funcs ((p . i),(p . (i + 1))) by A1, A2;
hence  q . i is set ; ::_thesis: verum
end;
end;

registration
cluster empty Relation-like Function-like FinSequence-like  ->  FuncSeq-like for set ;
coherence
 for b1 being FinSequence st b1 is empty holds
b1 is FuncSeq-like
proof
let p be FinSequence; ::_thesis: ( p is empty implies p is FuncSeq-like )
assume A1: p is empty ; ::_thesis: p is FuncSeq-like
take q = <*{}*>; :: according to FUNCT_7:def_8 ::_thesis: ( len q = (len p) + 1 & ( for i being Element of NAT st i in dom p holds
p . i in Funcs ((q . i),(q . (i + 1))) ) )
thus  len q = (len p) + 1 by A1, CARD_1:27, FINSEQ_1:40; ::_thesis: for i being Element of NAT st i in dom p holds
p . i in Funcs ((q . i),(q . (i + 1)))
thus  for i being Element of NAT st i in dom p holds
p . i in Funcs ((q . i),(q . (i + 1))) by A1; ::_thesis: verum
end;
end;

registration
let f be Function;
cluster<*f*> ->  FuncSeq-like ;
coherence
 <*f*> is FuncSeq-like
proof
take q = <*(dom f),(rng f)*>; :: according to FUNCT_7:def_8 ::_thesis: ( len q = (len <*f*>) + 1 & ( for i being Element of NAT st i in dom <*f*> holds
<*f*> . i in Funcs ((q . i),(q . (i + 1))) ) )
set p = <*f*>;
thus  len q = 1 + 1 by FINSEQ_1:44
.= (len <*f*>) + 1 by FINSEQ_1:40 ; ::_thesis: for i being Element of NAT st i in dom <*f*> holds
<*f*> . i in Funcs ((q . i),(q . (i + 1)))
let i be Element of NAT ; ::_thesis: ( i in dom <*f*> implies <*f*> . i in Funcs ((q . i),(q . (i + 1))) )
assume  i in dom <*f*> ; ::_thesis: <*f*> . i in Funcs ((q . i),(q . (i + 1)))
then  i in {1} by FINSEQ_1:2, FINSEQ_1:38;
then A1: i = 1 by TARSKI:def_1;
then A2: q . (i + 1) = rng f by FINSEQ_1:44;
 ( <*f*> . i = f & q . i = dom f ) by A1, FINSEQ_1:40, FINSEQ_1:44;
hence  <*f*> . i in Funcs ((q . i),(q . (i + 1))) by A2, FUNCT_2:def_2; ::_thesis: verum
end;
end;

registration
cluster non empty Relation-like non-empty NAT -defined Function-like finite FinSequence-like FinSubsequence-like FuncSeq-like for set ;
existence
 ex b1 being FinSequence st
( b1 is FuncSeq-like & not b1 is empty & b1 is non-empty )
proof
set f = the non empty Function;
take p = <* the non empty Function*>; ::_thesis: ( p is FuncSeq-like & not p is empty & p is non-empty )
thus  p is FuncSeq-like ; ::_thesis: ( not p is empty & p is non-empty )
thus  not p is empty ; ::_thesis: p is non-empty
let x be set ; :: according to FUNCT_1:def_9 ::_thesis: ( not x in proj1 p or not p . x is empty )
assume  x in dom p ; ::_thesis: not p . x is empty
then  x in {1} by FINSEQ_1:2, FINSEQ_1:38;
then  x = 1 by TARSKI:def_1;
hence  not p . x is empty by FINSEQ_1:40; ::_thesis: verum
end;
end;

definition
mode FuncSequence is FuncSeq-like FinSequence;
end;

theorem Th61: :: FUNCT_7:61
for X being set
for p being FuncSequence st p <> {} holds
dom (compose (p,X)) = (firstdom p) /\ X
proof
let X be set ; ::_thesis: for p being FuncSequence st p <> {} holds
dom (compose (p,X)) = (firstdom p) /\ X
defpred S1[ Function-yielding FinSequence] means ( $1 is FuncSequence & $1 <> {} implies dom (compose ($1,X)) = (firstdom $1) /\ X );
A1: for p being Function-yielding FinSequence st S1[p] holds
for f being Function holds S1[p ^ <*f*>]
proof
let q be Function-yielding FinSequence; ::_thesis: ( S1[q] implies for f being Function holds S1[q ^ <*f*>] )
assume A2: ( q is FuncSequence & q <> {} implies dom (compose (q,X)) = (firstdom q) /\ X ) ; ::_thesis: for f being Function holds S1[q ^ <*f*>]
let f be Function; ::_thesis: S1[q ^ <*f*>]
assume that
A3: q ^ <*f*> is FuncSequence and
 q ^ <*f*> <> {} ; ::_thesis: dom (compose ((q ^ <*f*>),X)) = (firstdom (q ^ <*f*>)) /\ X
A4: compose ((q ^ <*f*>),X) = f * (compose (q,X)) by Th41;
percases ( q = {} or q <> {} ) ;
suppose q = {} ; ::_thesis: dom (compose ((q ^ <*f*>),X)) = (firstdom (q ^ <*f*>)) /\ X
then A5: q ^ <*f*> = <*f*> by FINSEQ_1:34;
then A6: compose ((q ^ <*f*>),X) = f * (id X) by Th45;
 <*f*> ^ {} = <*f*> by FINSEQ_1:34;
then  firstdom (q ^ <*f*>) = dom f by A5, Th58;
hence  dom (compose ((q ^ <*f*>),X)) = (firstdom (q ^ <*f*>)) /\ X by A6, FUNCT_1:19; ::_thesis: verum
end;
supposeA7: q <> {} ; ::_thesis: dom (compose ((q ^ <*f*>),X)) = (firstdom (q ^ <*f*>)) /\ X
then consider y being set , s being FinSequence such that
A8: q = <*y*> ^ s and
A9: len q = (len s) + 1 by REWRITE1:5;
 q . 1 = y by A8, FINSEQ_1:41;
then A10: firstdom q = proj1 y by A8, Def6;
A11: len q >= 1 by A9, NAT_1:11;
then  len q in dom q by FINSEQ_3:25;
then A12: (q ^ <*f*>) . (len q) = q . (len q) by FINSEQ_1:def_7;
A13: rng (compose (q,X)) c= lastrng q by A7, Th59;
consider p being FinSequence such that
 len p = (len (q ^ <*f*>)) + 1 and
A14: for i being Element of NAT st i in dom (q ^ <*f*>) holds
(q ^ <*f*>) . i in Funcs ((p . i),(p . (i + 1))) by A3, Def8;
consider r being FinSequence, x being set  such that
A15: q = r ^ <*x*> by A7, FINSEQ_1:46;
 len <*f*> = 1 by FINSEQ_1:40;
then A16: len (q ^ <*f*>) = (len q) + 1 by FINSEQ_1:22;
then  len q <= len (q ^ <*f*>) by NAT_1:11;
then  len q in dom (q ^ <*f*>) by A11, FINSEQ_3:25;
then A17: (q ^ <*f*>) . (len q) in Funcs ((p . (len q)),(p . ((len q) + 1))) by A14;
 len <*x*> = 1 by FINSEQ_1:40;
then  len q = (len r) + 1 by A15, FINSEQ_1:22;
then  q . (len q) = x by A15, FINSEQ_1:42;
then consider g being Function such that
A18: x = g and
 dom g = p . (len q) and
A19: rng g c= p . ((len q) + 1) by A17, A12, FUNCT_2:def_2;
A20: (<*y*> ^ (s ^ <*f*>)) . 1 = y by FINSEQ_1:41;
 (len q) + 1 >= 1 by NAT_1:11;
then  (len q) + 1 in dom (q ^ <*f*>) by A16, FINSEQ_3:25;
then A21: (q ^ <*f*>) . ((len q) + 1) in Funcs ((p . ((len q) + 1)),(p . (((len q) + 1) + 1))) by A14;
 (q ^ <*f*>) . ((len q) + 1) = f by FINSEQ_1:42;
then A22: ex g being Function st
( f = g & dom g = p . ((len q) + 1) & rng g c= p . (((len q) + 1) + 1) ) by A21, FUNCT_2:def_2;
 q ^ <*f*> = <*y*> ^ (s ^ <*f*>) by A8, FINSEQ_1:32;
then A23: firstdom (q ^ <*f*>) = proj1 y by A20, Def6;
 lastrng q = rng g by A15, A18, Th58;
hence  dom (compose ((q ^ <*f*>),X)) = (firstdom (q ^ <*f*>)) /\ X by A2, A3, A4, A7, A23, A10, A19, A22, A13, Th60, RELAT_1:27, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
A24: S1[ {} ] ;
 for p being Function-yielding FinSequence holds S1[p] from FUNCT_7:sch_5(A24, A1);
hence  for p being FuncSequence st p <> {} holds
dom (compose (p,X)) = (firstdom p) /\ X ; ::_thesis: verum
end;

theorem Th62: :: FUNCT_7:62
for p being FuncSequence holds dom (compose (p,(firstdom p))) = firstdom p
proof
let p be FuncSequence; ::_thesis: dom (compose (p,(firstdom p))) = firstdom p
percases ( p = {} or p <> {} ) ;
suppose p = {} ; ::_thesis: dom (compose (p,(firstdom p))) = firstdom p
then  compose (p,(firstdom p)) = id (firstdom p) by Th39;
hence  dom (compose (p,(firstdom p))) = firstdom p by RELAT_1:45; ::_thesis: verum
end;
suppose p <> {} ; ::_thesis: dom (compose (p,(firstdom p))) = firstdom p
then  dom (compose (p,(firstdom p))) = (firstdom p) /\ (firstdom p) by Th61;
hence  dom (compose (p,(firstdom p))) = firstdom p ; ::_thesis: verum
end;
end;
end;

theorem :: FUNCT_7:63
for p being FuncSequence
for f being Function st rng f c= firstdom p holds
<*f*> ^ p is FuncSequence
proof
let p be FuncSequence; ::_thesis: for f being Function st rng f c= firstdom p holds
<*f*> ^ p is FuncSequence
let f be Function; ::_thesis: ( rng f c= firstdom p implies <*f*> ^ p is FuncSequence )
assume A1: rng f c= firstdom p ; ::_thesis: <*f*> ^ p is FuncSequence
consider q being FinSequence such that
A2: len q = (len p) + 1 and
A3: for i being Element of NAT st i in dom p holds
p . i in Funcs ((q . i),(q . (i + 1))) by Def8;
set r = <*(dom f)*> ^ q;
A4: len <*(dom f)*> = 1 by FINSEQ_1:40;
A5: len <*f*> = 1 by FINSEQ_1:40;
then A6: len (<*f*> ^ p) = (len p) + 1 by FINSEQ_1:22;
A7: now__::_thesis:_(_p_<>_{}_implies_rng_f_c=_q_._1_)
assume A8: p <> {} ; ::_thesis: rng f c= q . 1
then  len p >= 0 + 1 by NAT_1:13;
then  1 in dom p by FINSEQ_3:25;
then  p . 1 in Funcs ((q . 1),(q . (1 + 1))) by A3;
then  ex g being Function st
( p . 1 = g & dom g = q . 1 & rng g c= q . 2 ) by FUNCT_2:def_2;
hence  rng f c= q . 1 by A1, A8, Def6; ::_thesis: verum
end;
 <*f*> ^ p is FuncSeq-like
proof
take  <*(dom f)*> ^ q ; :: according to FUNCT_7:def_8 ::_thesis: ( len (<*(dom f)*> ^ q) = (len (<*f*> ^ p)) + 1 & ( for i being Element of NAT st i in dom (<*f*> ^ p) holds
(<*f*> ^ p) . i in Funcs (((<*(dom f)*> ^ q) . i),((<*(dom f)*> ^ q) . (i + 1))) ) )
 1 <= len q by A2, NAT_1:11;
then A9: 1 in dom q by FINSEQ_3:25;
thus  len (<*(dom f)*> ^ q) = (len (<*f*> ^ p)) + 1 by A2, A4, A6, FINSEQ_1:22; ::_thesis: for i being Element of NAT st i in dom (<*f*> ^ p) holds
(<*f*> ^ p) . i in Funcs (((<*(dom f)*> ^ q) . i),((<*(dom f)*> ^ q) . (i + 1)))
let i be Element of NAT ; ::_thesis: ( i in dom (<*f*> ^ p) implies (<*f*> ^ p) . i in Funcs (((<*(dom f)*> ^ q) . i),((<*(dom f)*> ^ q) . (i + 1))) )
assume A10: i in dom (<*f*> ^ p) ; ::_thesis: (<*f*> ^ p) . i in Funcs (((<*(dom f)*> ^ q) . i),((<*(dom f)*> ^ q) . (i + 1)))
then A11: i >= 1 by FINSEQ_3:25;
 {} c= q . 1 by XBOOLE_1:2;
then A12: rng f c= q . 1 by A1, A7, Th57;
A13: i <= (len p) + 1 by A6, A10, FINSEQ_3:25;
percases ( i = 1 or i <> 1 ) ;
supposeA14: i = 1 ; ::_thesis: (<*f*> ^ p) . i in Funcs (((<*(dom f)*> ^ q) . i),((<*(dom f)*> ^ q) . (i + 1)))
then A15: ( (<*(dom f)*> ^ q) . i = dom f & (<*f*> ^ p) . i = f ) by FINSEQ_1:41;
 (<*(dom f)*> ^ q) . (i + 1) = q . 1 by A4, A9, A14, FINSEQ_1:def_7;
hence  (<*f*> ^ p) . i in Funcs (((<*(dom f)*> ^ q) . i),((<*(dom f)*> ^ q) . (i + 1))) by A12, A15, FUNCT_2:def_2; ::_thesis: verum
end;
suppose i <> 1 ; ::_thesis: (<*f*> ^ p) . i in Funcs (((<*(dom f)*> ^ q) . i),((<*(dom f)*> ^ q) . (i + 1)))
then  i > 1 by A11, XXREAL_0:1;
then  i >= 1 + 1 by NAT_1:13;
then consider j being Nat such that
A16: i = (1 + 1) + j by NAT_1:10;
A17: i = (j + 1) + 1 by A16;
then  ( j + 1 >= 1 & j + 1 <= len p ) by A13, NAT_1:11, XREAL_1:6;
then A18: j + 1 in dom p by FINSEQ_3:25;
then A19: p . (j + 1) = (<*f*> ^ p) . i by A5, A17, FINSEQ_1:def_7;
A20: (j + 1) + 1 in dom q by A2, A18, Th22;
A21: p . (j + 1) in Funcs ((q . (j + 1)),(q . ((j + 1) + 1))) by A3, A18;
 j + 1 in dom q by A2, A18, Th22;
then  (<*(dom f)*> ^ q) . i = q . (j + 1) by A4, A16, A21, FINSEQ_1:def_7;
hence  (<*f*> ^ p) . i in Funcs (((<*(dom f)*> ^ q) . i),((<*(dom f)*> ^ q) . (i + 1))) by A4, A16, A21, A20, A19, FINSEQ_1:def_7; ::_thesis: verum
end;
end;
end;
hence  <*f*> ^ p is FuncSequence ; ::_thesis: verum
end;

theorem :: FUNCT_7:64
for p being FuncSequence
for f being Function st lastrng p c= dom f holds
p ^ <*f*> is FuncSequence
proof
let p be FuncSequence; ::_thesis: for f being Function st lastrng p c= dom f holds
p ^ <*f*> is FuncSequence
let f be Function; ::_thesis: ( lastrng p c= dom f implies p ^ <*f*> is FuncSequence )
assume A1: lastrng p c= dom f ; ::_thesis: p ^ <*f*> is FuncSequence
consider q being FinSequence such that
A2: len q = (len p) + 1 and
A3: for i being Element of NAT st i in dom p holds
p . i in Funcs ((q . i),(q . (i + 1))) by Def8;
consider q9 being FinSequence, x being set  such that
A4: q = q9 ^ <*x*> by A2, CARD_1:27, FINSEQ_1:46;
A5: now__::_thesis:_(_len_p_in_dom_p_implies_p_._(len_p)_in_Funcs_((q_._(len_p)),(dom_f))_)
assume A6: len p in dom p ; ::_thesis: p . (len p) in Funcs ((q . (len p)),(dom f))
then  p . (len p) in Funcs ((q . (len p)),(q . ((len p) + 1))) by A3;
then consider g being Function such that
A7: p . (len p) = g and
A8: dom g = q . (len p) and
 rng g c= q . ((len p) + 1) by FUNCT_2:def_2;
 lastrng p = rng g by A6, A7, Def7, RELAT_1:38;
hence  p . (len p) in Funcs ((q . (len p)),(dom f)) by A1, A7, A8, FUNCT_2:def_2; ::_thesis: verum
end;
A9: <*(dom f),(rng f)*> . 1 = dom f by FINSEQ_1:44;
A10: <*(dom f),(rng f)*> . 2 = rng f by FINSEQ_1:44;
 len <*f*> = 1 by FINSEQ_1:40;
then A11: len (p ^ <*f*>) = (len p) + 1 by FINSEQ_1:22;
set r = q9 ^ <*(dom f),(rng f)*>;
 len <*(dom f),(rng f)*> = 2 by FINSEQ_1:44;
then A12: len (q9 ^ <*(dom f),(rng f)*>) = (len q9) + (1 + 1) by FINSEQ_1:22;
A13: dom <*(dom f),(rng f)*> = Seg 2 by FINSEQ_1:89;
then A14: 1 in dom <*(dom f),(rng f)*> by FINSEQ_1:2, TARSKI:def_2;
 len <*x*> = 1 by FINSEQ_1:40;
then A15: len q = (len q9) + 1 by A4, FINSEQ_1:22;
A16: 2 in dom <*(dom f),(rng f)*> by A13, FINSEQ_1:2, TARSKI:def_2;
 p ^ <*f*> is FuncSeq-like
proof
take  q9 ^ <*(dom f),(rng f)*> ; :: according to FUNCT_7:def_8 ::_thesis: ( len (q9 ^ <*(dom f),(rng f)*>) = (len (p ^ <*f*>)) + 1 & ( for i being Element of NAT st i in dom (p ^ <*f*>) holds
(p ^ <*f*>) . i in Funcs (((q9 ^ <*(dom f),(rng f)*>) . i),((q9 ^ <*(dom f),(rng f)*>) . (i + 1))) ) )
thus  len (q9 ^ <*(dom f),(rng f)*>) = (len (p ^ <*f*>)) + 1 by A2, A15, A11, A12; ::_thesis: for i being Element of NAT st i in dom (p ^ <*f*>) holds
(p ^ <*f*>) . i in Funcs (((q9 ^ <*(dom f),(rng f)*>) . i),((q9 ^ <*(dom f),(rng f)*>) . (i + 1)))
let i be Element of NAT ; ::_thesis: ( i in dom (p ^ <*f*>) implies (p ^ <*f*>) . i in Funcs (((q9 ^ <*(dom f),(rng f)*>) . i),((q9 ^ <*(dom f),(rng f)*>) . (i + 1))) )
assume A17: i in dom (p ^ <*f*>) ; ::_thesis: (p ^ <*f*>) . i in Funcs (((q9 ^ <*(dom f),(rng f)*>) . i),((q9 ^ <*(dom f),(rng f)*>) . (i + 1)))
then A18: i >= 1 by FINSEQ_3:25;
 i <= (len p) + 1 by A11, A17, FINSEQ_3:25;
then A19: ( i = (len p) + 1 or len p >= i ) by NAT_1:8;
percases ( i = (len p) + 1 or i = len p or i < len p ) by A19, XXREAL_0:1;
supposeA20: i = (len p) + 1 ; ::_thesis: (p ^ <*f*>) . i in Funcs (((q9 ^ <*(dom f),(rng f)*>) . i),((q9 ^ <*(dom f),(rng f)*>) . (i + 1)))
then A21: ( (q9 ^ <*(dom f),(rng f)*>) . i = dom f & (p ^ <*f*>) . i = f ) by A2, A15, A14, A9, FINSEQ_1:42, FINSEQ_1:def_7;
 (q9 ^ <*(dom f),(rng f)*>) . (i + 1) = rng f by A2, A15, A12, A16, A10, A20, FINSEQ_1:def_7;
hence  (p ^ <*f*>) . i in Funcs (((q9 ^ <*(dom f),(rng f)*>) . i),((q9 ^ <*(dom f),(rng f)*>) . (i + 1))) by A21, FUNCT_2:def_2; ::_thesis: verum
end;
supposeA22: i = len p ; ::_thesis: (p ^ <*f*>) . i in Funcs (((q9 ^ <*(dom f),(rng f)*>) . i),((q9 ^ <*(dom f),(rng f)*>) . (i + 1)))
then  i in dom q9 by A2, A15, A18, FINSEQ_3:25;
then A23: ( (q9 ^ <*(dom f),(rng f)*>) . i = q9 . i & q9 . i = q . i ) by A4, FINSEQ_1:def_7;
 ( len p in dom p & (q9 ^ <*(dom f),(rng f)*>) . (i + 1) = dom f ) by A2, A15, A14, A9, A18, A22, FINSEQ_1:def_7, FINSEQ_3:25;
hence  (p ^ <*f*>) . i in Funcs (((q9 ^ <*(dom f),(rng f)*>) . i),((q9 ^ <*(dom f),(rng f)*>) . (i + 1))) by A5, A22, A23, FINSEQ_1:def_7; ::_thesis: verum
end;
supposeA24: i < len p ; ::_thesis: (p ^ <*f*>) . i in Funcs (((q9 ^ <*(dom f),(rng f)*>) . i),((q9 ^ <*(dom f),(rng f)*>) . (i + 1)))
then  i in dom q9 by A2, A15, A18, FINSEQ_3:25;
then A25: ( q . i = q9 . i & q9 . i = (q9 ^ <*(dom f),(rng f)*>) . i ) by A4, FINSEQ_1:def_7;
A26: i + 1 >= 1 by NAT_1:11;
 i in dom p by A18, A24, FINSEQ_3:25;
then A27: ( p . i in Funcs ((q . i),(q . (i + 1))) & p . i = (p ^ <*f*>) . i ) by A3, FINSEQ_1:def_7;
 i + 1 <= len p by A24, NAT_1:13;
then A28: i + 1 in dom q9 by A2, A15, A26, FINSEQ_3:25;
then  q . (i + 1) = q9 . (i + 1) by A4, FINSEQ_1:def_7;
hence  (p ^ <*f*>) . i in Funcs (((q9 ^ <*(dom f),(rng f)*>) . i),((q9 ^ <*(dom f),(rng f)*>) . (i + 1))) by A28, A25, A27, FINSEQ_1:def_7; ::_thesis: verum
end;
end;
end;
hence  p ^ <*f*> is FuncSequence ; ::_thesis: verum
end;

theorem :: FUNCT_7:65
for x, X being set
for p being FuncSequence st x in firstdom p & x in X holds
(apply (p,x)) . ((len p) + 1) = (compose (p,X)) . x
proof
let x, X be set ; ::_thesis: for p being FuncSequence st x in firstdom p & x in X holds
(apply (p,x)) . ((len p) + 1) = (compose (p,X)) . x
defpred S1[ Function-yielding FinSequence] means ( $1 is FuncSequence & x in firstdom $1 & x in X implies (apply ($1,x)) . ((len $1) + 1) = (compose ($1,X)) . x );
A1: for p being Function-yielding FinSequence st S1[p] holds
for f being Function holds S1[p ^ <*f*>]
proof
A2: dom (id X) = X ;
let p be Function-yielding FinSequence; ::_thesis: ( S1[p] implies for f being Function holds S1[p ^ <*f*>] )
assume A3: ( p is FuncSequence & x in firstdom p & x in X implies (apply (p,x)) . ((len p) + 1) = (compose (p,X)) . x ) ; ::_thesis: for f being Function holds S1[p ^ <*f*>]
let f be Function; ::_thesis: S1[p ^ <*f*>]
assume that
A4: p ^ <*f*> is FuncSequence and
A5: x in firstdom (p ^ <*f*>) and
A6: x in X ; ::_thesis: (apply ((p ^ <*f*>),x)) . ((len (p ^ <*f*>)) + 1) = (compose ((p ^ <*f*>),X)) . x
A7: p is FuncSequence by A4, Th60;
 (id X) . x = x by A6, FUNCT_1:17;
then A8: (f * (id X)) . x = f . x by A6, A2, FUNCT_1:13;
A9: len <*f*> = 1 by FINSEQ_1:40;
A10: compose ((p ^ <*f*>),X) = f * (compose (p,X)) by Th41;
A11: ( apply (<*f*>,x) = <*x,(f . x)*> & compose (<*f*>,X) = f * (id X) ) by Th45, Th47;
A12: apply ((p ^ <*f*>),x) = (apply (p,x)) ^ <*(f . ((apply (p,x)) . ((len p) + 1)))*> by Th42;
percases ( p = {} or p <> {} ) ;
suppose p = {} ; ::_thesis: (apply ((p ^ <*f*>),x)) . ((len (p ^ <*f*>)) + 1) = (compose ((p ^ <*f*>),X)) . x
then  p ^ <*f*> = <*f*> by FINSEQ_1:34;
hence  (apply ((p ^ <*f*>),x)) . ((len (p ^ <*f*>)) + 1) = (compose ((p ^ <*f*>),X)) . x by A9, A11, A8, FINSEQ_1:44; ::_thesis: verum
end;
supposeA13: p <> {} ; ::_thesis: (apply ((p ^ <*f*>),x)) . ((len (p ^ <*f*>)) + 1) = (compose ((p ^ <*f*>),X)) . x
then A14: dom (compose (p,X)) = (firstdom p) /\ X by A7, Th61;
A15: firstdom (p ^ <*f*>) = proj1 ((p ^ <*f*>) . 1) by Def6;
A16: firstdom p = proj1 (p . 1) by A13, Def6;
 len p >= 0 + 1 by A13, NAT_1:13;
then A17: 1 in dom p by FINSEQ_3:25;
then  p . 1 = (p ^ <*f*>) . 1 by FINSEQ_1:def_7;
then A18: x in (firstdom p) /\ X by A5, A6, A16, A15, XBOOLE_0:def_4;
 ( len (apply (p,x)) = (len p) + 1 & len (p ^ <*f*>) = (len p) + 1 ) by A9, Def5, FINSEQ_1:22;
hence (apply ((p ^ <*f*>),x)) . ((len (p ^ <*f*>)) + 1) = f . ((compose (p,X)) . x) by A3, A4, A5, A6, A12, A16, A15, A17, Th60, FINSEQ_1:42, FINSEQ_1:def_7
.= (compose ((p ^ <*f*>),X)) . x by A10, A14, A18, FUNCT_1:13 ;
::_thesis: verum
end;
end;
end;
A19: S1[ {} ] by Def6;
 for p being Function-yielding FinSequence holds S1[p] from FUNCT_7:sch_5(A19, A1);
hence  for p being FuncSequence st x in firstdom p & x in X holds
(apply (p,x)) . ((len p) + 1) = (compose (p,X)) . x ; ::_thesis: verum
end;

definition
let X, Y be set ;
assume X1: ( Y is empty implies X is empty ) ;
mode FuncSequence of X,Y ->  FuncSequence means :Def9: :: FUNCT_7:def 9
( firstdom it = X & lastrng it c= Y );
existence
 ex b1 being FuncSequence st
( firstdom b1 = X & lastrng b1 c= Y )
proof
set f = the Function of X,Y;
take p = <* the Function of X,Y*>; ::_thesis: ( firstdom p = X & lastrng p c= Y )
A1: ( p ^ {} = p & {} ^ p = p ) by FINSEQ_1:34;
 ( dom the Function of X,Y = X & rng the Function of X,Y c= Y ) by X1, FUNCT_2:def_1;
hence  ( firstdom p = X & lastrng p c= Y ) by A1, Th58; ::_thesis: verum
end;
end;

:: deftheorem Def9 defines FuncSequence FUNCT_7:def_9_:_
 for X, Y being set st ( Y is empty implies X is empty ) holds
for b3 being FuncSequence holds
( b3 is FuncSequence of X,Y iff ( firstdom b3 = X & lastrng b3 c= Y ) );

definition
let Y be non empty set ;
let X be set ;
let F be FuncSequence of X,Y;
:: original: compose
redefine func compose (F,X) ->  Function of X,Y;
coherence
 compose (F,X) is Function of X,Y
proof
A1: firstdom F = X by Def9;
now__::_thesis:_(_F_=_{}_implies_rng_(compose_(F,X))_=_{}_)
assume  F = {} ; ::_thesis: rng (compose (F,X)) = {}
then  ( X = {} & compose (F,X) = id X ) by A1, Def6, Th39;
hence  rng (compose (F,X)) = {} ; ::_thesis: verum
end;
then A2: rng (compose (F,X)) c= lastrng F by Th59, XBOOLE_1:2;
 lastrng F c= Y by Def9;
then A3: rng (compose (F,X)) c= Y by A2, XBOOLE_1:1;
 dom (compose (F,X)) = X by A1, Th62;
hence  compose (F,X) is Function of X,Y by A3, FUNCT_2:def_1, RELSET_1:4; ::_thesis: verum
end;
end;

definition
let q be non empty non-empty FinSequence;
mode FuncSequence of q ->  FinSequence means :Def10: :: FUNCT_7:def 10
( (len it) + 1 = len q & ( for i being Element of NAT st i in dom it holds
it . i in Funcs ((q . i),(q . (i + 1))) ) );
existence
 ex b1 being FinSequence st
( (len b1) + 1 = len q & ( for i being Element of NAT st i in dom b1 holds
b1 . i in Funcs ((q . i),(q . (i + 1))) ) )
proof
defpred S1[ set , set ] means  ex i being Element of NAT st
( i = $1 & $2 in Funcs ((q . i),(q . (i + 1))) );
consider n being Nat such that
A1: len q = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A2: for x being set st x in Seg n holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in Seg n implies ex y being set st S1[x,y] )
assume A3: x in Seg n ; ::_thesis: ex y being set st S1[x,y]
then reconsider i = x as Element of NAT ;
A4: i <= n by A3, FINSEQ_1:1;
then  ( i + 1 >= 1 & i + 1 <= n + 1 ) by NAT_1:11, XREAL_1:6;
then A5: i + 1 in dom q by A1, FINSEQ_3:25;
 n <= n + 1 by NAT_1:11;
then A6: i <= n + 1 by A4, XXREAL_0:2;
 i >= 1 by A3, FINSEQ_1:1;
then  i in dom q by A1, A6, FINSEQ_3:25;
then reconsider X = q . i, Y = q . (i + 1) as non empty set by A5;
set y = the Function of X,Y;
take  the Function of X,Y ; ::_thesis: S1[x, the Function of X,Y]
take  i ; ::_thesis: ( i = x & the Function of X,Y in Funcs ((q . i),(q . (i + 1))) )
thus  ( i = x & the Function of X,Y in Funcs ((q . i),(q . (i + 1))) ) by FUNCT_2:8; ::_thesis: verum
end;
consider f being Function such that
A7: dom f = Seg n and
A8: for x being set st x in Seg n holds
S1[x,f . x] from CLASSES1:sch_1(A2);
reconsider f = f as FinSequence by A7, FINSEQ_1:def_2;
take  f ; ::_thesis: ( (len f) + 1 = len q & ( for i being Element of NAT st i in dom f holds
f . i in Funcs ((q . i),(q . (i + 1))) ) )
thus  (len f) + 1 = len q by A1, A7, FINSEQ_1:def_3; ::_thesis: for i being Element of NAT st i in dom f holds
f . i in Funcs ((q . i),(q . (i + 1)))
let i be Element of NAT ; ::_thesis: ( i in dom f implies f . i in Funcs ((q . i),(q . (i + 1))) )
assume  i in dom f ; ::_thesis: f . i in Funcs ((q . i),(q . (i + 1)))
then  ex j being Element of NAT st
( j = i & f . i in Funcs ((q . j),(q . (j + 1))) ) by A7, A8;
hence  f . i in Funcs ((q . i),(q . (i + 1))) ; ::_thesis: verum
end;
end;

:: deftheorem Def10 defines FuncSequence FUNCT_7:def_10_:_
 for q being non empty non-empty FinSequence
for b2 being FinSequence holds
( b2 is FuncSequence of q iff ( (len b2) + 1 = len q & ( for i being Element of NAT st i in dom b2 holds
b2 . i in Funcs ((q . i),(q . (i + 1))) ) ) );

registration
let q be non empty non-empty FinSequence;
cluster  ->  non-empty FuncSeq-like for FuncSequence of q;
coherence
 for b1 being FuncSequence of q holds
( b1 is FuncSeq-like & b1 is non-empty )
proof
let p be FuncSequence of q; ::_thesis: ( p is FuncSeq-like & p is non-empty )
thus  p is FuncSeq-like ::_thesis: p is non-empty
proof
take  q ; :: according to FUNCT_7:def_8 ::_thesis: ( len q = (len p) + 1 & ( for i being Element of NAT st i in dom p holds
p . i in Funcs ((q . i),(q . (i + 1))) ) )
thus  ( len q = (len p) + 1 & ( for i being Element of NAT st i in dom p holds
p . i in Funcs ((q . i),(q . (i + 1))) ) ) by Def10; ::_thesis: verum
end;
let x be set ; :: according to FUNCT_1:def_9 ::_thesis: ( not x in proj1 p or not p . x is empty )
assume A1: x in dom p ; ::_thesis: not p . x is empty
then reconsider i = x as Element of NAT ;
 len q = (len p) + 1 by Def10;
then  ( i in dom q & i + 1 in dom q ) by A1, Th22;
then reconsider X = q . i, Y = q . (i + 1) as non empty set ;
 p . i in Funcs (X,Y) by A1, Def10;
then  ex f being Function st
( p . x = f & dom f = X & rng f c= Y ) by FUNCT_2:def_2;
hence  not p . x is empty ; ::_thesis: verum
end;
end;

theorem Th66: :: FUNCT_7:66
for q being non empty non-empty FinSequence
for p being FuncSequence of q st p <> {} holds
( firstdom p = q . 1 & lastrng p c= q . (len q) )
proof
let q be non empty non-empty FinSequence; ::_thesis: for p being FuncSequence of q st p <> {} holds
( firstdom p = q . 1 & lastrng p c= q . (len q) )
let p be FuncSequence of q; ::_thesis: ( p <> {} implies ( firstdom p = q . 1 & lastrng p c= q . (len q) ) )
assume A1: p <> {} ; ::_thesis: ( firstdom p = q . 1 & lastrng p c= q . (len q) )
then  1 in dom p by FINSEQ_5:6;
then  p . 1 in Funcs ((q . 1),(q . (1 + 1))) by Def10;
then  ex f being Function st
( p . 1 = f & dom f = q . 1 & rng f c= q . 2 ) by FUNCT_2:def_2;
hence  firstdom p = q . 1 by A1, Def6; ::_thesis: lastrng p c= q . (len q)
 len p in dom p by A1, FINSEQ_5:6;
then  p . (len p) in Funcs ((q . (len p)),(q . ((len p) + 1))) by Def10;
then A2: ex g being Function st
( p . (len p) = g & dom g = q . (len p) & rng g c= q . ((len p) + 1) ) by FUNCT_2:def_2;
 (len p) + 1 = len q by Def10;
hence  lastrng p c= q . (len q) by A1, A2, Def7; ::_thesis: verum
end;

theorem :: FUNCT_7:67
for q being non empty non-empty FinSequence
for p being FuncSequence of q holds
( dom (compose (p,(q . 1))) = q . 1 & rng (compose (p,(q . 1))) c= q . (len q) )
proof
let q be non empty non-empty FinSequence; ::_thesis: for p being FuncSequence of q holds
( dom (compose (p,(q . 1))) = q . 1 & rng (compose (p,(q . 1))) c= q . (len q) )
let p be FuncSequence of q; ::_thesis: ( dom (compose (p,(q . 1))) = q . 1 & rng (compose (p,(q . 1))) c= q . (len q) )
percases ( p = {} or p <> {} ) ;
supposeA1: p = {} ; ::_thesis: ( dom (compose (p,(q . 1))) = q . 1 & rng (compose (p,(q . 1))) c= q . (len q) )
A2: len q = (len p) + 1 by Def10;
 ( compose (p,(q . 1)) = id (q . 1) & len p = 0 ) by A1, Th39;
hence  ( dom (compose (p,(q . 1))) = q . 1 & rng (compose (p,(q . 1))) c= q . (len q) ) by A2, RELAT_1:45; ::_thesis: verum
end;
supposeA3: p <> {} ; ::_thesis: ( dom (compose (p,(q . 1))) = q . 1 & rng (compose (p,(q . 1))) c= q . (len q) )
then A4: lastrng p c= q . (len q) by Th66;
 ( firstdom p = q . 1 & rng (compose (p,(q . 1))) c= lastrng p ) by A3, Th59, Th66;
hence  ( dom (compose (p,(q . 1))) = q . 1 & rng (compose (p,(q . 1))) c= q . (len q) ) by A4, Th62, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;

registration
let X be set ;
let f be Function of NAT,(bool [:X,X:]);
let n be Nat;
clusterf . n ->  Relation-like ;
coherence
 f . n is Relation-like
proof
reconsider n = n as Element of NAT by ORDINAL1:def_12;
 f . n is Element of bool [:X,X:] ;
hence  f . n is Relation-like ; ::_thesis: verum
end;
end;

Lm1:  for X being set
for f being Function of X,X holds rng f c= dom f
proof
let X be set ; ::_thesis: for f being Function of X,X holds rng f c= dom f
let f be Function of X,X; ::_thesis: rng f c= dom f
 dom f = X by FUNCT_2:52;
hence  rng f c= dom f ; ::_thesis: verum
end;

Lm2:  for f being Relation
for n being Element of NAT
for p1, p2 being Function of NAT,(bool [:(field f),(field f):]) st p1 . 0 = id (field f) & ( for k being Nat holds p1 . (k + 1) = f * (p1 . k) ) & p2 . 0 = id (field f) & ( for k being Nat holds p2 . (k + 1) = f * (p2 . k) ) holds
p1 = p2
proof
let f be Relation; ::_thesis: for n being Element of NAT
for p1, p2 being Function of NAT,(bool [:(field f),(field f):]) st p1 . 0 = id (field f) & ( for k being Nat holds p1 . (k + 1) = f * (p1 . k) ) & p2 . 0 = id (field f) & ( for k being Nat holds p2 . (k + 1) = f * (p2 . k) ) holds
p1 = p2
let n be Element of NAT ; ::_thesis: for p1, p2 being Function of NAT,(bool [:(field f),(field f):]) st p1 . 0 = id (field f) & ( for k being Nat holds p1 . (k + 1) = f * (p1 . k) ) & p2 . 0 = id (field f) & ( for k being Nat holds p2 . (k + 1) = f * (p2 . k) ) holds
p1 = p2
let p1, p2 be Function of NAT,(bool [:(field f),(field f):]); ::_thesis: ( p1 . 0 = id (field f) & ( for k being Nat holds p1 . (k + 1) = f * (p1 . k) ) & p2 . 0 = id (field f) & ( for k being Nat holds p2 . (k + 1) = f * (p2 . k) ) implies p1 = p2 )
assume that
A1: p1 . 0 = id (field f) and
A2: for k being Nat holds p1 . (k + 1) = f * (p1 . k) and
A3: p2 . 0 = id (field f) and
A4: for k being Nat holds p2 . (k + 1) = f * (p2 . k) ; ::_thesis: p1 = p2
defpred S1[ Nat, Relation, set ] means $3 = f * $2;
A5: for k being Nat holds S1[k,p1 . k,p1 . (k + 1)] by A2;
set FX = bool [:(field f),(field f):];
reconsider ID = id (field f) as Element of bool [:(field f),(field f):] ;
A6: p2 . 0 = ID by A3;
A7: for k being Nat
for x, y1, y2 being Element of bool [:(field f),(field f):] st S1[k,x,y1] & S1[k,x,y2] holds
y1 = y2 ;
A8: for k being Nat holds S1[k,p2 . k,p2 . (k + 1)] by A4;
A9: p1 . 0 = ID by A1;
 p1 = p2 from NAT_1:sch_14(A9, A5, A6, A8, A7);
hence  p1 = p2 ; ::_thesis: verum
end;

definition
let f be Relation;
let n be Nat;
func iter (f,n) ->  Relation means :Def11: :: FUNCT_7:def 11
 ex p being Function of NAT,(bool [:(field f),(field f):]) st
( it = p . n & p . 0 = id (field f) & ( for k being Nat holds p . (k + 1) = f * (p . k) ) );
existence
 ex b1 being Relation ex p being Function of NAT,(bool [:(field f),(field f):]) st
( b1 = p . n & p . 0 = id (field f) & ( for k being Nat holds p . (k + 1) = f * (p . k) ) )
proof
reconsider n9 = n as Element of NAT by ORDINAL1:def_12;
defpred S1[ Nat, Relation, set ] means $3 = f * $2;
set FX = bool [:(field f),(field f):];
reconsider ID = id (field f) as Element of bool [:(field f),(field f):] ;
A1: for n being Element of NAT
for x being Element of bool [:(field f),(field f):] ex y being Element of bool [:(field f),(field f):] st S1[n,x,y]
proof
 ( dom f c= field f & rng f c= field f ) by XBOOLE_1:7;
then reconsider h = f as Relation of (field f),(field f) by RELSET_1:4;
let n be Element of NAT ; ::_thesis: for x being Element of bool [:(field f),(field f):] ex y being Element of bool [:(field f),(field f):] st S1[n,x,y]
let x be Element of bool [:(field f),(field f):]; ::_thesis: ex y being Element of bool [:(field f),(field f):] st S1[n,x,y]
reconsider g = x as Relation of (field f),(field f) ;
 h * g is Element of bool [:(field f),(field f):] ;
hence  ex y being Element of bool [:(field f),(field f):] st S1[n,x,y] ; ::_thesis: verum
end;
consider p being Function of NAT,(bool [:(field f),(field f):]) such that
A2: ( p . 0 = ID & ( for n being Element of NAT holds S1[n,p . n,p . (n + 1)] ) ) from RECDEF_1:sch_2(A1);
A3: for n being Nat holds S1[n,p . n,p . (n + 1)]
proof
let n be Nat; ::_thesis: S1[n,p . n,p . (n + 1)]
 n in NAT by ORDINAL1:def_12;
hence  S1[n,p . n,p . (n + 1)] by A2; ::_thesis: verum
end;
 p . n9 is Relation of (field f),(field f) ;
hence  ex b1 being Relation ex p being Function of NAT,(bool [:(field f),(field f):]) st
( b1 = p . n & p . 0 = id (field f) & ( for k being Nat holds p . (k + 1) = f * (p . k) ) ) by A2, A3; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Relation st ex p being Function of NAT,(bool [:(field f),(field f):]) st
( b1 = p . n & p . 0 = id (field f) & ( for k being Nat holds p . (k + 1) = f * (p . k) ) ) & ex p being Function of NAT,(bool [:(field f),(field f):]) st
( b2 = p . n & p . 0 = id (field f) & ( for k being Nat holds p . (k + 1) = f * (p . k) ) ) holds
b1 = b2 by Lm2;
end;

:: deftheorem Def11 defines iter FUNCT_7:def_11_:_
 for f being Relation
for n being Nat
for b3 being Relation holds
( b3 = iter (f,n) iff ex p being Function of NAT,(bool [:(field f),(field f):]) st
( b3 = p . n & p . 0 = id (field f) & ( for k being Nat holds p . (k + 1) = f * (p . k) ) ) );

registration
let f be Function;
let n be Nat;
cluster iter (f,n) ->  Function-like ;
coherence
 iter (f,n) is Function-like
proof
consider p being Function of NAT,(bool [:(field f),(field f):]) such that
A1: ( p . n = iter (f,n) & p . 0 = id (field f) ) and
A2: for k being Nat holds p . (k + 1) = f * (p . k) by Def11;
defpred S1[ Nat] means p . f is Function;
A3: S1[ 0 ] by A1;
A4: for m being Nat st S1[m] holds
S1[m + 1]
proof
let m be Nat; ::_thesis: ( S1[m] implies S1[m + 1] )
assume  S1[m] ; ::_thesis: S1[m + 1]
then reconsider g = p . m as Function ;
 p . (m + 1) = g * f by A2;
hence  S1[m + 1] ; ::_thesis: verum
end;
 for m being Nat holds S1[m] from NAT_1:sch_2(A3, A4);
hence  iter (f,n) is Function-like by A1; ::_thesis: verum
end;
end;

    Lm3:  for R being Relation holds
( (id (field R)) * R = R & R * (id (field R)) = R )
proof
let R be Relation; ::_thesis: ( (id (field R)) * R = R & R * (id (field R)) = R )
 ( dom R c= field R & rng R c= field R ) by XBOOLE_1:7;
hence  ( (id (field R)) * R = R & R * (id (field R)) = R ) by RELAT_1:51, RELAT_1:53; ::_thesis: verum
end;

theorem Th68: :: FUNCT_7:68
for R being Relation holds iter (R,0) = id (field R)
proof
let R be Relation; ::_thesis: iter (R,0) = id (field R)
 ex p being Function of NAT,(bool [:(field R),(field R):]) st
( iter (R,0) = p . 0 & p . 0 = id (field R) & ( for k being Nat holds p . (k + 1) = R * (p . k) ) ) by Def11;
hence  iter (R,0) = id (field R) ; ::_thesis: verum
end;

Lm4:  for R being Relation st rng R c= dom R holds
iter (R,0) = id (dom R)
proof
let R be Relation; ::_thesis: ( rng R c= dom R implies iter (R,0) = id (dom R) )
assume  rng R c= dom R ; ::_thesis: iter (R,0) = id (dom R)
then  field R = dom R by XBOOLE_1:12;
hence  iter (R,0) = id (dom R) by Th68; ::_thesis: verum
end;

theorem Th69: :: FUNCT_7:69
for R being Relation
for n being Nat holds iter (R,(n + 1)) = R * (iter (R,n))
proof
let R be Relation; ::_thesis: for n being Nat holds iter (R,(n + 1)) = R * (iter (R,n))
let n be Nat; ::_thesis: iter (R,(n + 1)) = R * (iter (R,n))
consider p being Function of NAT,(bool [:(field R),(field R):]) such that
A1: ( p . (n + 1) = iter (R,(n + 1)) & p . 0 = id (field R) ) and
A2: for k being Nat holds p . (k + 1) = R * (p . k) by Def11;
 p . (n + 1) = R * (p . n) by A2;
hence  iter (R,(n + 1)) = R * (iter (R,n)) by A1, A2, Def11; ::_thesis: verum
end;

theorem Th70: :: FUNCT_7:70
for R being Relation holds iter (R,1) = R
proof
let R be Relation; ::_thesis: iter (R,1) = R
thus  iter (R,1) = iter (R,(0 + 1))
.= R * (iter (R,0)) by Th69
.= R * (id (field R)) by Th68
.= R by Lm3 ; ::_thesis: verum
end;

theorem Th71: :: FUNCT_7:71
for R being Relation
for n being Nat holds iter (R,(n + 1)) = (iter (R,n)) * R
proof
let R be Relation; ::_thesis: for n being Nat holds iter (R,(n + 1)) = (iter (R,n)) * R
let n be Nat; ::_thesis: iter (R,(n + 1)) = (iter (R,n)) * R
defpred S1[ Nat] means  iter (R,($1 + 1)) = (iter (R,$1)) * R;
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: iter (R,(k + 1)) = (iter (R,k)) * R ; ::_thesis: S1[k + 1]
thus (iter (R,(k + 1))) * R = (R * (iter (R,k))) * R by Th69
.= R * ((iter (R,k)) * R) by RELAT_1:36
.= iter (R,((k + 1) + 1)) by A2, Th69 ; ::_thesis: verum
end;
iter (R,(0 + 1)) = R by Th70
.= (id (field R)) * R by Lm3
.= (iter (R,0)) * R by Th68 ;
then A3: S1[ 0 ] ;
 for k being Nat holds S1[k] from NAT_1:sch_2(A3, A1);
hence  iter (R,(n + 1)) = (iter (R,n)) * R ; ::_thesis: verum
end;

theorem Th72: :: FUNCT_7:72
for n being Element of NAT
for R being Relation holds
( dom (iter (R,n)) c= field R & rng (iter (R,n)) c= field R )
proof
let n be Element of NAT ; ::_thesis: for R being Relation holds
( dom (iter (R,n)) c= field R & rng (iter (R,n)) c= field R )
let R be Relation; ::_thesis: ( dom (iter (R,n)) c= field R & rng (iter (R,n)) c= field R )
defpred S1[ Element of NAT ] means ( dom (iter (R,$1)) c= field R & rng (iter (R,$1)) c= field R );
A1: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
 iter (R,(k + 1)) = (iter (R,k)) * R by Th71;
then A2: dom (iter (R,(k + 1))) c= dom (iter (R,k)) by RELAT_1:25;
 iter (R,(k + 1)) = R * (iter (R,k)) by Th69;
then A3: rng (iter (R,(k + 1))) c= rng (iter (R,k)) by RELAT_1:26;
assume  ( dom (iter (R,k)) c= field R & rng (iter (R,k)) c= field R ) ; ::_thesis: S1[k + 1]
hence  S1[k + 1] by A2, A3, XBOOLE_1:1; ::_thesis: verum
end;
 iter (R,0) = id (field R) by Th68;
then A4: S1[ 0 ] by RELAT_1:45;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A4, A1);
hence  ( dom (iter (R,n)) c= field R & rng (iter (R,n)) c= field R ) ; ::_thesis: verum
end;

theorem :: FUNCT_7:73
for n being Element of NAT
for R being Relation st n <> 0 holds
( dom (iter (R,n)) c= dom R & rng (iter (R,n)) c= rng R )
proof
let n be Element of NAT ; ::_thesis: for R being Relation st n <> 0 holds
( dom (iter (R,n)) c= dom R & rng (iter (R,n)) c= rng R )
let R be Relation; ::_thesis: ( n <> 0 implies ( dom (iter (R,n)) c= dom R & rng (iter (R,n)) c= rng R ) )
defpred S1[ Nat] means ( dom (iter (R,($1 + 1))) c= dom R & rng (iter (R,($1 + 1))) c= rng R );
assume  n <> 0 ; ::_thesis: ( dom (iter (R,n)) c= dom R & rng (iter (R,n)) c= rng R )
then A1: ex k being Nat st n = k + 1 by NAT_1:6;
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 that
 dom (iter (R,(k + 1))) c= dom R and
 rng (iter (R,(k + 1))) c= rng R ; ::_thesis: S1[k + 1]
 ( iter (R,((k + 1) + 1)) = R * (iter (R,(k + 1))) & iter (R,((k + 1) + 1)) = (iter (R,(k + 1))) * R ) by Th69, Th71;
hence  S1[k + 1] by RELAT_1:25, RELAT_1:26; ::_thesis: verum
end;
A3: S1[ 0 ] by Th70;
 for k being Nat holds S1[k] from NAT_1:sch_2(A3, A2);
hence  ( dom (iter (R,n)) c= dom R & rng (iter (R,n)) c= rng R ) by A1; ::_thesis: verum
end;

theorem Th74: :: FUNCT_7:74
for R being Relation
for n being Nat st rng R c= dom R holds
( dom (iter (R,n)) = dom R & rng (iter (R,n)) c= dom R )
proof
let R be Relation; ::_thesis: for n being Nat st rng R c= dom R holds
( dom (iter (R,n)) = dom R & rng (iter (R,n)) c= dom R )
let n be Nat; ::_thesis: ( rng R c= dom R implies ( dom (iter (R,n)) = dom R & rng (iter (R,n)) c= dom R ) )
defpred S1[ Nat] means ( dom (iter (R,$1)) = dom R & rng (iter (R,$1)) c= dom R );
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: ( dom (iter (R,k)) = dom R & rng (iter (R,k)) c= dom R ) ; ::_thesis: S1[k + 1]
 iter (R,(k + 1)) = R * (iter (R,k)) by Th69;
then A3: rng (iter (R,(k + 1))) c= rng (iter (R,k)) by RELAT_1:26;
 iter (R,(k + 1)) = (iter (R,k)) * R by Th71;
hence  S1[k + 1] by A2, A3, RELAT_1:27, XBOOLE_1:1; ::_thesis: verum
end;
assume  rng R c= dom R ; ::_thesis: ( dom (iter (R,n)) = dom R & rng (iter (R,n)) c= dom R )
then  iter (R,0) = id (dom R) by Lm4;
then A4: S1[ 0 ] by RELAT_1:45;
 for k being Nat holds S1[k] from NAT_1:sch_2(A4, A1);
hence  ( dom (iter (R,n)) = dom R & rng (iter (R,n)) c= dom R ) ; ::_thesis: verum
end;

theorem Th75: :: FUNCT_7:75
for n being Element of NAT
for R being Relation holds (id (field R)) * (iter (R,n)) = iter (R,n)
proof
let n be Element of NAT ; ::_thesis: for R being Relation holds (id (field R)) * (iter (R,n)) = iter (R,n)
let R be Relation; ::_thesis: (id (field R)) * (iter (R,n)) = iter (R,n)
 dom (iter (R,n)) c= field R by Th72;
hence  (id (field R)) * (iter (R,n)) = iter (R,n) by RELAT_1:51; ::_thesis: verum
end;

theorem :: FUNCT_7:76
for n being Element of NAT
for R being Relation holds (iter (R,n)) * (id (field R)) = iter (R,n)
proof
let n be Element of NAT ; ::_thesis: for R being Relation holds (iter (R,n)) * (id (field R)) = iter (R,n)
let R be Relation; ::_thesis: (iter (R,n)) * (id (field R)) = iter (R,n)
 rng (iter (R,n)) c= field R by Th72;
hence  (iter (R,n)) * (id (field R)) = iter (R,n) by RELAT_1:53; ::_thesis: verum
end;

theorem Th77: :: FUNCT_7:77
for m, n being Element of NAT
for R being Relation holds (iter (R,m)) * (iter (R,n)) = iter (R,(n + m))
proof
let m, n be Element of NAT ; ::_thesis: for R being Relation holds (iter (R,m)) * (iter (R,n)) = iter (R,(n + m))
let R be Relation; ::_thesis: (iter (R,m)) * (iter (R,n)) = iter (R,(n + m))
defpred S1[ Element of NAT ] means (iter (R,$1)) * (iter (R,n)) = iter (R,(n + $1));
A1: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A2: (iter (R,k)) * (iter (R,n)) = iter (R,(n + k)) ; ::_thesis: S1[k + 1]
thus (iter (R,(k + 1))) * (iter (R,n)) = (R * (iter (R,k))) * (iter (R,n)) by Th69
.= R * ((iter (R,k)) * (iter (R,n))) by RELAT_1:36
.= iter (R,((n + k) + 1)) by A2, Th69
.= iter (R,(n + (k + 1))) ; ::_thesis: verum
end;
(iter (R,0)) * (iter (R,n)) = (id (field R)) * (iter (R,n)) by Th68
.= iter (R,(n + 0)) by Th75 ;
then A3: S1[ 0 ] ;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A3, A1);
hence  (iter (R,m)) * (iter (R,n)) = iter (R,(n + m)) ; ::_thesis: verum
end;

Lm5:  for m, k being Element of NAT
for R being Relation st iter ((iter (R,m)),k) = iter (R,(m * k)) holds
iter ((iter (R,m)),(k + 1)) = iter (R,(m * (k + 1)))
proof
let m, k be Element of NAT ; ::_thesis: for R being Relation st iter ((iter (R,m)),k) = iter (R,(m * k)) holds
iter ((iter (R,m)),(k + 1)) = iter (R,(m * (k + 1)))
let R be Relation; ::_thesis: ( iter ((iter (R,m)),k) = iter (R,(m * k)) implies iter ((iter (R,m)),(k + 1)) = iter (R,(m * (k + 1))) )
assume A1: iter ((iter (R,m)),k) = iter (R,(m * k)) ; ::_thesis: iter ((iter (R,m)),(k + 1)) = iter (R,(m * (k + 1)))
thus  iter ((iter (R,m)),(k + 1)) = (iter (R,m)) * (iter ((iter (R,m)),k)) by Th69
.= iter (R,((m * k) + (m * 1))) by A1, Th77
.= iter (R,(m * (k + 1))) ; ::_thesis: verum
end;

theorem :: FUNCT_7:78
for n, m being Element of NAT
for R being Relation st n <> 0 holds
iter ((iter (R,m)),n) = iter (R,(m * n))
proof
let n, m be Element of NAT ; ::_thesis: for R being Relation st n <> 0 holds
iter ((iter (R,m)),n) = iter (R,(m * n))
let R be Relation; ::_thesis: ( n <> 0 implies iter ((iter (R,m)),n) = iter (R,(m * n)) )
defpred S1[ Element of NAT ] means  iter ((iter (R,m)),($1 + 1)) = iter (R,(m * ($1 + 1)));
A1: for k being Element of NAT st S1[k] holds
S1[k + 1] by Lm5;
A2: S1[ 0 ] by Th70;
A3: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A2, A1);
assume  n <> 0 ; ::_thesis: iter ((iter (R,m)),n) = iter (R,(m * n))
then consider k being Nat such that
A4: n = k + 1 by NAT_1:6;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
 n = k + 1 by A4;
hence  iter ((iter (R,m)),n) = iter (R,(m * n)) by A3; ::_thesis: verum
end;

theorem Th79: :: FUNCT_7:79
for m, n being Element of NAT
for R being Relation st rng R c= dom R holds
iter ((iter (R,m)),n) = iter (R,(m * n))
proof
let m, n be Element of NAT ; ::_thesis: for R being Relation st rng R c= dom R holds
iter ((iter (R,m)),n) = iter (R,(m * n))
let R be Relation; ::_thesis: ( rng R c= dom R implies iter ((iter (R,m)),n) = iter (R,(m * n)) )
defpred S1[ Element of NAT ] means  iter ((iter (R,m)),$1) = iter (R,(m * $1));
assume A1: rng R c= dom R ; ::_thesis: iter ((iter (R,m)),n) = iter (R,(m * n))
then  dom (iter (R,m)) = dom R by Th74;
then  field (iter (R,m)) = dom R by A1, Th74, XBOOLE_1:12;
then  iter ((iter (R,m)),0) = id (dom R) by Th68
.= id (field R) by A1, XBOOLE_1:12
.= iter (R,(m * 0)) by Th68 ;
then A2: S1[ 0 ] ;
A3: for k being Element of NAT st S1[k] holds
S1[k + 1] by Lm5;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A2, A3);
hence  iter ((iter (R,m)),n) = iter (R,(m * n)) ; ::_thesis: verum
end;

theorem :: FUNCT_7:80
for n being Element of NAT holds iter ({},n) = {}
proof
let n be Element of NAT ; ::_thesis: iter ({},n) = {}
defpred S1[ Element of NAT ] means  iter ({},$1) = {} ;
A1: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume  iter ({},k) = {} ; ::_thesis: S1[k + 1]
thus  iter ({},(k + 1)) = (iter ({},k)) * {} by Th69
.= {} ; ::_thesis: verum
end;
iter ({},0) = id (field {}) by Th68
.= {} ;
then A2: S1[ 0 ] ;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A2, A1);
hence  iter ({},n) = {} ; ::_thesis: verum
end;

theorem :: FUNCT_7:81
for X being set
for n being Element of NAT holds iter ((id X),n) = id X
proof
let X be set ; ::_thesis: for n being Element of NAT holds iter ((id X),n) = id X
let n be Element of NAT ; ::_thesis: iter ((id X),n) = id X
defpred S1[ Element of NAT ] means  iter ((id X),$1) = id X;
A1: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A2: S1[k] ; ::_thesis: S1[k + 1]
thus  iter ((id X),(k + 1)) = (iter ((id X),k)) * (id X) by Th69
.= id X by A2, FUNCT_2:17 ; ::_thesis: verum
end;
 id (field (id X)) = id X ;
then A3: S1[ 0 ] by Th68;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A3, A1);
hence  iter ((id X),n) = id X ; ::_thesis: verum
end;

theorem :: FUNCT_7:82
for R being Relation st rng R misses dom R holds
iter (R,2) = {}
proof
let R be Relation; ::_thesis: ( rng R misses dom R implies iter (R,2) = {} )
assume A1: rng R misses dom R ; ::_thesis: iter (R,2) = {}
thus  iter (R,2) = iter (R,(1 + 1))
.= (iter (R,1)) * R by Th71
.= R * R by Th70
.= {} by A1, RELAT_1:44 ; ::_thesis: verum
end;

theorem :: FUNCT_7:83
for X being set
for n being Nat
for f being Function of X,X holds iter (f,n) is Function of X,X
proof
let X be set ; ::_thesis: for n being Nat
for f being Function of X,X holds iter (f,n) is Function of X,X
let n be Nat; ::_thesis: for f being Function of X,X holds iter (f,n) is Function of X,X
let f be Function of X,X; ::_thesis: iter (f,n) is Function of X,X
A1: ( X = {} implies X = {} ) ;
then A2: dom f = X by FUNCT_2:def_1;
A3: rng f c= X ;
then  ( dom (iter (f,n)) = X & rng (iter (f,n)) c= X ) by A2, Th74;
then reconsider R = iter (f,n) as Relation of X,X by RELSET_1:4;
 dom R = X by A2, A3, Th74;
hence  iter (f,n) is Function of X,X by A1, FUNCT_2:def_1; ::_thesis: verum
end;

theorem :: FUNCT_7:84
for X being set
for f being Function of X,X holds iter (f,0) = id X
proof
let X be set ; ::_thesis: for f being Function of X,X holds iter (f,0) = id X
let f be Function of X,X; ::_thesis: iter (f,0) = id X
 ( iter (f,0) = id (field f) & field f = dom f ) by Lm1, Th68, XBOOLE_1:12;
hence  iter (f,0) = id X by FUNCT_2:52; ::_thesis: verum
end;

theorem :: FUNCT_7:85
for X being set
for m, n being Element of NAT
for f being Function of X,X holds iter ((iter (f,m)),n) = iter (f,(m * n))
proof
let X be set ; ::_thesis: for m, n being Element of NAT
for f being Function of X,X holds iter ((iter (f,m)),n) = iter (f,(m * n))
let m, n be Element of NAT ; ::_thesis: for f being Function of X,X holds iter ((iter (f,m)),n) = iter (f,(m * n))
let f be Function of X,X; ::_thesis: iter ((iter (f,m)),n) = iter (f,(m * n))
 rng f c= dom f by Lm1;
hence  iter ((iter (f,m)),n) = iter (f,(m * n)) by Th79; ::_thesis: verum
end;

theorem :: FUNCT_7:86
for X being set
for n being Element of NAT
for f being PartFunc of X,X holds iter (f,n) is PartFunc of X,X
proof
let X be set ; ::_thesis: for n being Element of NAT
for f being PartFunc of X,X holds iter (f,n) is PartFunc of X,X
let n be Element of NAT ; ::_thesis: for f being PartFunc of X,X holds iter (f,n) is PartFunc of X,X
let f be PartFunc of X,X; ::_thesis: iter (f,n) is PartFunc of X,X
A1: field f = (dom f) \/ (rng f) ;
 rng (iter (f,n)) c= field f by Th72;
then A2: rng (iter (f,n)) c= X by A1, XBOOLE_1:1;
 dom (iter (f,n)) c= field f by Th72;
then  dom (iter (f,n)) c= X by A1, XBOOLE_1:1;
hence  iter (f,n) is PartFunc of X,X by A2, RELSET_1:4; ::_thesis: verum
end;

theorem :: FUNCT_7:87
for a, X being set
for f being Function
for n being Element of NAT st n <> 0 & a in X & f = X --> a holds
iter (f,n) = f
proof
let a, X be set ; ::_thesis: for f being Function
for n being Element of NAT st n <> 0 & a in X & f = X --> a holds
iter (f,n) = f
let f be Function; ::_thesis: for n being Element of NAT st n <> 0 & a in X & f = X --> a holds
iter (f,n) = f
let n be Element of NAT ; ::_thesis: ( n <> 0 & a in X & f = X --> a implies iter (f,n) = f )
assume that
A1: n <> 0 and
A2: a in X and
A3: f = X --> a ; ::_thesis: iter (f,n) = f
defpred S1[ Element of NAT ] means  iter (f,($1 + 1)) = f;
A4: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
A5: dom f = X by A3, FUNCOP_1:13;
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A6: S1[k] ; ::_thesis: S1[k + 1]
A7: now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
(iter_(f,((k_+_1)_+_1)))_._x_=_f_._x
let x be set ; ::_thesis: ( x in dom f implies (iter (f,((k + 1) + 1))) . x = f . x )
assume A8: x in dom f ; ::_thesis: (iter (f,((k + 1) + 1))) . x = f . x
then A9: f . x = a by A3, A5, FUNCOP_1:7;
thus (iter (f,((k + 1) + 1))) . x = (f * f) . x by A6, Th71
.= f . (f . x) by A8, FUNCT_1:13
.= f . x by A2, A3, A9, FUNCOP_1:7 ; ::_thesis: verum
end;
 rng f = {a} by A2, A3, FUNCOP_1:8;
then  rng f c= dom f by A2, A5, ZFMISC_1:31;
then  dom (iter (f,((k + 1) + 1))) = dom f by Th74;
hence  S1[k + 1] by A7, FUNCT_1:2; ::_thesis: verum
end;
A10: S1[ 0 ] by Th70;
A11: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A10, A4);
consider k being Nat such that
A12: n = k + 1 by A1, NAT_1:6;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
 n = k + 1 by A12;
hence  iter (f,n) = f by A11; ::_thesis: verum
end;

theorem :: FUNCT_7:88
for f being Function
for n being Element of NAT holds iter (f,n) = compose ((n |-> f),(field f))
proof
let f be Function; ::_thesis: for n being Element of NAT holds iter (f,n) = compose ((n |-> f),(field f))
defpred S1[ Element of NAT ] means  iter (f,$1) = compose (($1 |-> f),(field f));
A1: now__::_thesis:_for_n_being_Element_of_NAT_st_S1[n]_holds_
S1[n_+_1]
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume  S1[n] ; ::_thesis: S1[n + 1]
then  iter (f,(n + 1)) = f * (compose ((n |-> f),(field f))) by Th71
.= compose (((n |-> f) ^ <*f*>),(field f)) by Th41
.= compose (((n + 1) |-> f),(field f)) by FINSEQ_2:60 ;
hence  S1[n + 1] ; ::_thesis: verum
end;
iter (f,0) = id (field f) by Th68
.= compose ({},(field f)) by Th39
.= compose ((0 |-> f),(field f)) ;
then A2: S1[ 0 ] ;
thus  for n being Element of NAT holds S1[n] from NAT_1:sch_1(A2, A1); ::_thesis: verum
end;

begin

theorem :: FUNCT_7:89
for f, g being Function
for x, y being set st g c= f & not x in dom g holds
g c= f +* (x,y)
proof
let f, g be Function; ::_thesis: for x, y being set st g c= f & not x in dom g holds
g c= f +* (x,y)
let x, y be set ; ::_thesis: ( g c= f & not x in dom g implies g c= f +* (x,y) )
assume that
A1: g c= f and
A2: not x in dom g ; ::_thesis: g c= f +* (x,y)
A3: now__::_thesis:_for_z_being_set_st_z_in_dom_g_holds_
g_._z_=_(f_+*_(x,y))_._z
let z be set ; ::_thesis: ( z in dom g implies g . z = (f +* (x,y)) . z )
assume A4: z in dom g ; ::_thesis: g . z = (f +* (x,y)) . z
hence g . z = f . z by A1, GRFUNC_1:2
.= (f +* (x,y)) . z by A2, A4, Th32 ;
::_thesis: verum
end;
 dom g c= dom f by A1, GRFUNC_1:2;
then  dom g c= dom (f +* (x,y)) by Th30;
hence  g c= f +* (x,y) by A3, GRFUNC_1:2; ::_thesis: verum
end;

theorem :: FUNCT_7:90
for f, g being Function
for A being set st f | A = g | A & f,g equal_outside A holds
f = g
proof
let f, g be Function; ::_thesis: for A being set st f | A = g | A & f,g equal_outside A holds
f = g
let A be set ; ::_thesis: ( f | A = g | A & f,g equal_outside A implies f = g )
assume A1: ( f | A = g | A & f,g equal_outside A ) ; ::_thesis: f = g
thus f = f | ((dom f) \/ A) by RELAT_1:68, XBOOLE_1:7
.= f | (((dom f) \ A) \/ A) by XBOOLE_1:39
.= (f | ((dom f) \ A)) \/ (f | A) by RELAT_1:78
.= (g | ((dom g) \ A)) \/ (g | A) by A1, Def2
.= g | (((dom g) \ A) \/ A) by RELAT_1:78
.= g | ((dom g) \/ A) by XBOOLE_1:39
.= g by RELAT_1:68, XBOOLE_1:7 ; ::_thesis: verum
end;

theorem :: FUNCT_7:91
for f being Function
for a, b, A being set st a in A holds
f,f +* (a,b) equal_outside A
proof
let f be Function; ::_thesis: for a, b, A being set st a in A holds
f,f +* (a,b) equal_outside A
let a, b, A be set ; ::_thesis: ( a in A implies f,f +* (a,b) equal_outside A )
percases ( a in dom f or not a in dom f ) ;
supposeA1: a in dom f ; ::_thesis: ( a in A implies f,f +* (a,b) equal_outside A )
assume  a in A ; ::_thesis: f,f +* (a,b) equal_outside A
then  {a} c= A by ZFMISC_1:31;
then A2: dom (a .--> b) c= A by FUNCOP_1:13;
 f +* (a,b) = f +* (a .--> b) by A1, Def3;
hence  f,f +* (a,b) equal_outside A by A2, Th29; ::_thesis: verum
end;
suppose not a in dom f ; ::_thesis: ( a in A implies f,f +* (a,b) equal_outside A )
then  f +* (a,b) = f by Def3;
hence  ( a in A implies f,f +* (a,b) equal_outside A ) by Th25; ::_thesis: verum
end;
end;
end;

theorem Th92: :: FUNCT_7:92
for f being Function
for a, b, A being set holds
( a in A or (f +* (a,b)) | A = f | A )
proof
let f be Function; ::_thesis: for a, b, A being set holds
( a in A or (f +* (a,b)) | A = f | A )
let a, b, A be set ; ::_thesis: ( a in A or (f +* (a,b)) | A = f | A )
percases ( a in dom f or not a in dom f ) ;
supposeA1: a in dom f ; ::_thesis: ( a in A or (f +* (a,b)) | A = f | A )
assume  not a in A ; ::_thesis: (f +* (a,b)) | A = f | A
then  {a} misses A by ZFMISC_1:50;
then A2: dom (a .--> b) misses A by FUNCOP_1:13;
thus (f +* (a,b)) | A = (f +* (a .--> b)) | A by A1, Def3
.= f | A by A2, FUNCT_4:72 ; ::_thesis: verum
end;
suppose not a in dom f ; ::_thesis: ( a in A or (f +* (a,b)) | A = f | A )
hence  ( a in A or (f +* (a,b)) | A = f | A ) by Def3; ::_thesis: verum
end;
end;
end;

theorem :: FUNCT_7:93
for f, g being Function
for a, b, A being set st f | A = g | A holds
(f +* (a,b)) | A = (g +* (a,b)) | A
proof
let f, g be Function; ::_thesis: for a, b, A being set st f | A = g | A holds
(f +* (a,b)) | A = (g +* (a,b)) | A
let a, b, A be set ; ::_thesis: ( f | A = g | A implies (f +* (a,b)) | A = (g +* (a,b)) | A )
assume A1: f | A = g | A ; ::_thesis: (f +* (a,b)) | A = (g +* (a,b)) | A
percases ( ( a in A & a in dom g ) or ( a in A & not a in dom g ) or not a in A ) ;
supposethat A2: a in A and
A3: a in dom g ; ::_thesis: (f +* (a,b)) | A = (g +* (a,b)) | A
now__::_thesis:_a_in_dom_f
assume  not a in dom f ; ::_thesis: contradiction
then  not a in (dom f) /\ A by XBOOLE_0:def_4;
then  not a in dom (g | A) by A1, RELAT_1:61;
then  not a in (dom g) /\ A by RELAT_1:61;
hence  contradiction by A2, A3, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (f +* (a,b)) | A = (f +* (a .--> b)) | A by Def3
.= (g | A) +* ((a .--> b) | A) by A1, FUNCT_4:71
.= (g +* (a .--> b)) | A by FUNCT_4:71
.= (g +* (a,b)) | A by A3, Def3 ;
::_thesis: verum
end;
supposethat A4: a in A and
A5: not a in dom g ; ::_thesis: (f +* (a,b)) | A = (g +* (a,b)) | A
now__::_thesis:_not_a_in_dom_f
assume  a in dom f ; ::_thesis: contradiction
then  a in (dom f) /\ A by A4, XBOOLE_0:def_4;
then  a in dom (g | A) by A1, RELAT_1:61;
then  a in (dom g) /\ A by RELAT_1:61;
hence  contradiction by A5, XBOOLE_0:def_4; ::_thesis: verum
end;
hence (f +* (a,b)) | A = g | A by A1, Def3
.= (g +* (a,b)) | A by A5, Def3 ;
::_thesis: verum
end;
supposeA6: not a in A ; ::_thesis: (f +* (a,b)) | A = (g +* (a,b)) | A
hence (f +* (a,b)) | A = f | A by Th92
.= (g +* (a,b)) | A by A1, A6, Th92 ;
::_thesis: verum
end;
end;
end;

theorem Th94: :: FUNCT_7:94
for f being Function
for a, b being set holds (f +* (a .--> b)) . a = b
proof
let f be Function; ::_thesis: for a, b being set holds (f +* (a .--> b)) . a = b
let a, b be set ; ::_thesis: (f +* (a .--> b)) . a = b
 dom (a .--> b) = {a} by FUNCOP_1:13;
then  a in dom (a .--> b) by TARSKI:def_1;
hence (f +* (a .--> b)) . a = (a .--> b) . a by FUNCT_4:13
.= b by FUNCOP_1:72 ;
::_thesis: verum
end;

theorem :: FUNCT_7:95
for a, b being set holds <*a*> +* (1,b) = <*b*>
proof
let a, b be set ; ::_thesis: <*a*> +* (1,b) = <*b*>
A1: dom <*b*> = {1} by FINSEQ_1:2, FINSEQ_1:def_8;
A2: dom <*a*> = {1} by FINSEQ_1:2, FINSEQ_1:def_8;
then  1 in dom <*a*> by TARSKI:def_1;
then A3: <*a*> +* (1,b) = <*a*> +* (1 .--> b) by Def3;
A4: for x being set st x in {1} holds
(<*a*> +* (1,b)) . x = <*b*> . x
proof
let x be set ; ::_thesis: ( x in {1} implies (<*a*> +* (1,b)) . x = <*b*> . x )
assume  x in {1} ; ::_thesis: (<*a*> +* (1,b)) . x = <*b*> . x
then A5: x = 1 by TARSKI:def_1;
hence (<*a*> +* (1,b)) . x = b by A3, Th94
.= <*b*> . x by A5, FINSEQ_1:def_8 ;
::_thesis: verum
end;
dom (<*a*> +* (1,b)) = (dom <*a*>) \/ (dom (1 .--> b)) by A3, FUNCT_4:def_1
.= {1} \/ {1} by A2, FUNCOP_1:13
.= {1} ;
hence  <*a*> +* (1,b) = <*b*> by A1, A4, FUNCT_1:2; ::_thesis: verum
end;

theorem :: FUNCT_7:96
for f being Function
for x being set st x in dom f holds
f +* (x .--> (f . x)) = f
proof
let f be Function; ::_thesis: for x being set st x in dom f holds
f +* (x .--> (f . x)) = f
let x be set ; ::_thesis: ( x in dom f implies f +* (x .--> (f . x)) = f )
assume  x in dom f ; ::_thesis: f +* (x .--> (f . x)) = f
hence f +* (x .--> (f . x)) = f +* (x,(f . x)) by Def3
.= f by Th35 ;
::_thesis: verum
end;


theorem Th97: :: FUNCT_7:97
for w being FinSequence
for r being set
for i being Nat holds len (w +* (i,r)) = len w
proof
let w be FinSequence; ::_thesis: for r being set
for i being Nat holds len (w +* (i,r)) = len w
let r be set ; ::_thesis: for i being Nat holds len (w +* (i,r)) = len w
let i be Nat; ::_thesis: len (w +* (i,r)) = len w
 dom (w +* (i,r)) = dom w by Th30;
then  Seg (len (w +* (i,r))) = dom w by FINSEQ_1:def_3
.= Seg (len w) by FINSEQ_1:def_3 ;
hence  len (w +* (i,r)) = len w by FINSEQ_1:6; ::_thesis: verum
end;

theorem :: FUNCT_7:98
for i being Nat
for D being non empty set
for w being FinSequence of D
for r being Element of D st i in dom w holds
w +* (i,r) = ((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)
proof
let i be Nat; ::_thesis: for D being non empty set
for w being FinSequence of D
for r being Element of D st i in dom w holds
w +* (i,r) = ((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)
let D be non empty set ; ::_thesis: for w being FinSequence of D
for r being Element of D st i in dom w holds
w +* (i,r) = ((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)
let w be FinSequence of D; ::_thesis: for r being Element of D st i in dom w holds
w +* (i,r) = ((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)
let r be Element of D; ::_thesis: ( i in dom w implies w +* (i,r) = ((w | (i -' 1)) ^ <*r*>) ^ (w /^ i) )
assume A1: i in dom w ; ::_thesis: w +* (i,r) = ((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)
then A2: 1 <= i by FINSEQ_3:25;
A3: i <= len w by A1, FINSEQ_3:25;
A4: len (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) = (len ((w | (i -' 1)) ^ <*r*>)) + (len (w /^ i)) by FINSEQ_1:22
.= ((len (w | (i -' 1))) + (len <*r*>)) + (len (w /^ i)) by FINSEQ_1:22
.= ((len (w | (i -' 1))) + 1) + (len (w /^ i)) by FINSEQ_1:39
.= ((i -' 1) + 1) + (len (w /^ i)) by A3, FINSEQ_1:59, NAT_D:44
.= ((i -' 1) + 1) + ((len w) - i) by A3, RFINSEQ:def_1
.= ((i - 1) + 1) + ((len w) - i) by A2, XREAL_1:233 ;
 for j being Nat st 1 <= j & j <= len (w +* (i,r)) holds
(w +* (i,r)) . j = (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j
proof
A5: len (w | (i -' 1)) = i -' 1 by A3, FINSEQ_1:59, NAT_D:44
.= i - 1 by A2, XREAL_1:233 ;
let j be Nat; ::_thesis: ( 1 <= j & j <= len (w +* (i,r)) implies (w +* (i,r)) . j = (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j )
assume that
A6: 1 <= j and
A7: j <= len (w +* (i,r)) ; ::_thesis: (w +* (i,r)) . j = (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j
A8: len ((w | (i -' 1)) ^ <*r*>) = (len (w | (i -' 1))) + (len <*r*>) by FINSEQ_1:22
.= (len (w | (i -' 1))) + 1 by FINSEQ_1:39
.= (i -' 1) + 1 by A3, FINSEQ_1:59, NAT_D:44
.= (i - 1) + 1 by A2, XREAL_1:233
.= i ;
A9: len (w +* (i,r)) = len w by Th97;
now__::_thesis:_(_(_i_<_j_&_(w_+*_(i,r))_._j_=_(((w_|_(i_-'_1))_^_<*r*>)_^_(w_/^_i))_._j_)_or_(_j_=_i_&_(w_+*_(i,r))_._j_=_(((w_|_(i_-'_1))_^_<*r*>)_^_(w_/^_i))_._j_)_or_(_j_<_i_&_(w_+*_(i,r))_._j_=_(((w_|_(i_-'_1))_^_<*r*>)_^_(w_/^_i))_._j_)_)
percases ( i < j or j = i or j < i ) by XXREAL_0:1;
caseA10: i < j ; ::_thesis: (w +* (i,r)) . j = (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j
then  i + 1 <= j by NAT_1:13;
then A11: (i + 1) - i <= j - i by XREAL_1:9;
then A12: 1 <= j -' i by NAT_D:39;
A13: j - i <= (len w) - i by A7, A9, XREAL_1:9;
A14: i <= len w by A1, FINSEQ_3:25;
len (w /^ i) = (len w) -' i by RFINSEQ:29
.= (len w) - i by A14, XREAL_1:233 ;
then  j -' i <= len (w /^ i) by A11, A13, NAT_D:39;
then A15: j -' i in dom (w /^ i) by A12, FINSEQ_3:25;
 j <= len (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) by A4, A7, Th97;
then (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j = (w /^ i) . (j - (len ((w | (i -' 1)) ^ <*r*>))) by A8, A10, FINSEQ_1:24
.= (w /^ i) . (j -' i) by A8, A10, XREAL_1:233
.= w . ((j -' i) + i) by A14, A15, RFINSEQ:def_1
.= w . j by A10, XREAL_1:235 ;
hence  (w +* (i,r)) . j = (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j by A10, Th32; ::_thesis: verum
end;
caseA16: j = i ; ::_thesis: (w +* (i,r)) . j = (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j
A17: i = (len (w | (i -' 1))) + 1 by A5;
(((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j = ((w | (i -' 1)) ^ <*r*>) . i by A6, A8, A16, FINSEQ_1:64
.= r by A17, FINSEQ_1:42 ;
hence  (w +* (i,r)) . j = (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j by A1, A16, Th31; ::_thesis: verum
end;
caseA18: j < i ; ::_thesis: (w +* (i,r)) . j = (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j
then  j + 1 <= i by NAT_1:13;
then A19: (j + 1) - 1 <= i - 1 by XREAL_1:9;
then A20: j <= i -' 1 by A2, XREAL_1:233;
(((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j = ((w | (i -' 1)) ^ <*r*>) . j by A6, A8, A18, FINSEQ_1:64
.= (w | (i -' 1)) . j by A6, A5, A19, FINSEQ_1:64
.= w . j by A20, FINSEQ_3:112 ;
hence  (w +* (i,r)) . j = (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j by A18, Th32; ::_thesis: verum
end;
end;
end;
hence  (w +* (i,r)) . j = (((w | (i -' 1)) ^ <*r*>) ^ (w /^ i)) . j ; ::_thesis: verum
end;
hence  w +* (i,r) = ((w | (i -' 1)) ^ <*r*>) ^ (w /^ i) by A4, Th97, FINSEQ_1:14; ::_thesis: verum
end;


definition
let F be Function;
let x, y be set ;
func Swap (F,x,y) ->  set  equals :Def12: :: FUNCT_7:def 12
(F +* (x,(F . y))) +* (y,(F . x)) if ( x in dom F & y in dom F )
otherwise F;
correctness
coherence
( ( x in dom F & y in dom F implies (F +* (x,(F . y))) +* (y,(F . x)) is set ) & ( ( not x in dom F or not y in dom F ) implies F is set ) );
consistency
for b1 being set holds verum;
;
end;

:: deftheorem Def12 defines Swap FUNCT_7:def_12_:_
 for F being Function
for x, y being set holds
( ( x in dom F & y in dom F implies Swap (F,x,y) = (F +* (x,(F . y))) +* (y,(F . x)) ) & ( ( not x in dom F or not y in dom F ) implies Swap (F,x,y) = F ) );

registration
let F be Function;
let x, y be set ;
cluster Swap (F,x,y) ->  Relation-like Function-like ;
coherence
 ( Swap (F,x,y) is Relation-like & Swap (F,x,y) is Function-like )
proof
percases ( ( x in dom F & y in dom F ) or not x in dom F or not y in dom F ) ;
suppose ( x in dom F & y in dom F ) ; ::_thesis: ( Swap (F,x,y) is Relation-like & Swap (F,x,y) is Function-like )
then  Swap (F,x,y) = (F +* (x,(F . y))) +* (y,(F . x)) by Def12;
hence  ( Swap (F,x,y) is Relation-like & Swap (F,x,y) is Function-like ) ; ::_thesis: verum
end;
suppose ( not x in dom F or not y in dom F ) ; ::_thesis: ( Swap (F,x,y) is Relation-like & Swap (F,x,y) is Function-like )
hence  ( Swap (F,x,y) is Relation-like & Swap (F,x,y) is Function-like ) by Def12; ::_thesis: verum
end;
end;
end;
end;

theorem Th99: :: FUNCT_7:99
for F being Function
for x, y being set holds dom (Swap (F,x,y)) = dom F
proof
let F be Function; ::_thesis: for x, y being set holds dom (Swap (F,x,y)) = dom F
let x, y be set ; ::_thesis: dom (Swap (F,x,y)) = dom F
percases ( ( x in dom F & y in dom F ) or not x in dom F or not y in dom F ) ;
suppose ( x in dom F & y in dom F ) ; ::_thesis: dom (Swap (F,x,y)) = dom F
hence  dom (Swap (F,x,y)) = dom ((F +* (x,(F . y))) +* (y,(F . x))) by Def12
.= dom (F +* (x,(F . y))) by Th30
.= dom F by Th30 ;
::_thesis: verum
end;
suppose ( not x in dom F or not y in dom F ) ; ::_thesis: dom (Swap (F,x,y)) = dom F
hence  dom (Swap (F,x,y)) = dom F by Def12; ::_thesis: verum
end;
end;
end;

theorem Th100: :: FUNCT_7:100
for F being Function
for x, y being set holds rng (F +* (x,y)) c= (rng F) \/ {y}
proof
let F be Function; ::_thesis: for x, y being set holds rng (F +* (x,y)) c= (rng F) \/ {y}
let x, y be set ; ::_thesis: rng (F +* (x,y)) c= (rng F) \/ {y}
A1: rng (x .--> y) = {y} by FUNCOP_1:8;
percases ( x in dom F or not x in dom F ) ;
suppose x in dom F ; ::_thesis: rng (F +* (x,y)) c= (rng F) \/ {y}
then  F +* (x,y) = F +* (x .--> y) by Def3;
hence  rng (F +* (x,y)) c= (rng F) \/ {y} by A1, FUNCT_4:17; ::_thesis: verum
end;
suppose not x in dom F ; ::_thesis: rng (F +* (x,y)) c= (rng F) \/ {y}
then  F +* (x,y) = F by Def3;
hence  rng (F +* (x,y)) c= (rng F) \/ {y} by XBOOLE_1:7; ::_thesis: verum
end;
end;
end;

theorem Th101: :: FUNCT_7:101
for F being Function
for x, y being set holds rng F c= (rng (F +* (x,y))) \/ {(F . x)}
proof
let F be Function; ::_thesis: for x, y being set holds rng F c= (rng (F +* (x,y))) \/ {(F . x)}
let x, y be set ; ::_thesis: rng F c= (rng (F +* (x,y))) \/ {(F . x)}
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng F or z in (rng (F +* (x,y))) \/ {(F . x)} )
assume  z in rng F ; ::_thesis: z in (rng (F +* (x,y))) \/ {(F . x)}
then consider e being set  such that
A1: e in dom F and
A2: z = F . e by FUNCT_1:def_3;
A3: dom F = dom (F +* (x,y)) by Th30;
percases ( e = x or e <> x ) ;
suppose e = x ; ::_thesis: z in (rng (F +* (x,y))) \/ {(F . x)}
then  z in {(F . x)} by A2, TARSKI:def_1;
hence  z in (rng (F +* (x,y))) \/ {(F . x)} by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose e <> x ; ::_thesis: z in (rng (F +* (x,y))) \/ {(F . x)}
then  (F +* (x,y)) . e = F . e by Th32;
then  z in rng (F +* (x,y)) by A1, A2, A3, FUNCT_1:3;
hence  z in (rng (F +* (x,y))) \/ {(F . x)} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;

theorem Th102: :: FUNCT_7:102
for F being Function
for x, y being set st x in dom F holds
y in rng (F +* (x,y))
proof
let F be Function; ::_thesis: for x, y being set st x in dom F holds
y in rng (F +* (x,y))
let x, y be set ; ::_thesis: ( x in dom F implies y in rng (F +* (x,y)) )
assume  x in dom F ; ::_thesis: y in rng (F +* (x,y))
then  ( x in dom (F +* (x,y)) & (F +* (x,y)) . x = y ) by Th30, Th31;
hence  y in rng (F +* (x,y)) by FUNCT_1:3; ::_thesis: verum
end;

theorem :: FUNCT_7:103
for F being Function
for x, y being set holds rng (Swap (F,x,y)) = rng F
proof
let F be Function; ::_thesis: for x, y being set holds rng (Swap (F,x,y)) = rng F
let x, y be set ; ::_thesis: rng (Swap (F,x,y)) = rng F
percases ( ( x in dom F & y in dom F ) or not x in dom F or not y in dom F ) ;
supposethat A1: x in dom F and
A2: y in dom F ; ::_thesis: rng (Swap (F,x,y)) = rng F
 F . x in rng F by A1, FUNCT_1:3;
then  F . x in (rng F) \/ {(F . y)} by XBOOLE_0:def_3;
then ((rng F) \/ {(F . y)}) \/ {(F . x)} = (rng F) \/ {(F . y)} by ZFMISC_1:40
.= rng F by A2, FUNCT_1:3, ZFMISC_1:40 ;
then A3: (rng (F +* (x,(F . y)))) \/ {(F . x)} c= rng F by Th100, XBOOLE_1:9;
A4: (F +* (x,(F . y))) . y = F . y
proof
percases ( x <> y or x = y ) ;
suppose x <> y ; ::_thesis: (F +* (x,(F . y))) . y = F . y
hence  (F +* (x,(F . y))) . y = F . y by Th32; ::_thesis: verum
end;
suppose x = y ; ::_thesis: (F +* (x,(F . y))) . y = F . y
hence  (F +* (x,(F . y))) . y = F . y by A1, Th31; ::_thesis: verum
end;
end;
end;
A5: rng F c= (rng (F +* (x,(F . y)))) \/ {(F . x)} by Th101;
A6: Swap (F,x,y) = (F +* (x,(F . y))) +* (y,(F . x)) by A1, A2, Def12;
then  rng (Swap (F,x,y)) c= (rng (F +* (x,(F . y)))) \/ {(F . x)} by Th100;
then A7: rng (Swap (F,x,y)) c= rng F by A3, XBOOLE_1:1;
A8: y in dom (F +* (x,(F . y))) by A2, Th30;
A9: F . y in rng (Swap (F,x,y))
proof
percases ( F . x = F . y or F . x <> F . y ) ;
suppose F . x = F . y ; ::_thesis: F . y in rng (Swap (F,x,y))
hence  F . y in rng (Swap (F,x,y)) by A6, A8, Th102; ::_thesis: verum
end;
supposeA10: F . x <> F . y ; ::_thesis: F . y in rng (Swap (F,x,y))
A11: dom (Swap (F,x,y)) = dom F by Th99;
(Swap (F,x,y)) . x = (F +* (x,(F . y))) . x by A6, A10, Th32
.= F . y by A1, Th31 ;
hence  F . y in rng (Swap (F,x,y)) by A1, A11, FUNCT_1:3; ::_thesis: verum
end;
end;
end;
 F . x in rng (Swap (F,x,y)) by A6, A8, Th102;
then  F . x in (rng (Swap (F,x,y))) \/ {(F . y)} by XBOOLE_0:def_3;
then ((rng (Swap (F,x,y))) \/ {(F . y)}) \/ {(F . x)} = (rng (Swap (F,x,y))) \/ {(F . y)} by ZFMISC_1:40
.= rng (Swap (F,x,y)) by A9, ZFMISC_1:40 ;
then  (rng (F +* (x,(F . y)))) \/ {(F . x)} c= rng (Swap (F,x,y)) by A6, A4, Th101, XBOOLE_1:9;
then  rng F c= rng (Swap (F,x,y)) by A5, XBOOLE_1:1;
hence  rng (Swap (F,x,y)) = rng F by A7, XBOOLE_0:def_10; ::_thesis: verum
end;
suppose ( not x in dom F or not y in dom F ) ; ::_thesis: rng (Swap (F,x,y)) = rng F
hence  rng (Swap (F,x,y)) = rng F by Def12; ::_thesis: verum
end;
end;
end;

scheme :: FUNCT_7:sch 6
Sch6{ F1() ->  set , F2() ->  non empty set , F3( set ) ->  set } :
 F1(), {  F3(d) where d is Element of F2() : d in F1()  }  are_equipotent
provided
A1: F1() c= F2() and
A2: for d1, d2 being Element of F2() st d1 in F1() & d2 in F1() & F3(d1) = F3(d2) holds
d1 = d2
proof
percases ( F1() <> {} or F1() = {} ) ;
suppose F1() <> {} ; ::_thesis: F1(), {  F3(d) where d is Element of F2() : d in F1()  }  are_equipotent
then reconsider D = F1() as non empty set ;
set X =  {  F3(d) where d is Element of D : d in F1()  }  ;
set Y =  {  F3(d) where d is Element of F2() : d in F1()  }  ;
A3: now__::_thesis:_for_x_being_set_holds_
(_(_x_in__{__F3(d)_where_d_is_Element_of_D_:_d_in_F1()__}__implies_x_in__{__F3(d)_where_d_is_Element_of_F2()_:_d_in_F1()__}__)_&_(_x_in__{__F3(d)_where_d_is_Element_of_F2()_:_d_in_F1()__}__implies_x_in__{__F3(d)_where_d_is_Element_of_D_:_d_in_F1()__}__)_)
let x be set ; ::_thesis: ( ( x in  {  F3(d) where d is Element of D : d in F1()  }  implies x in  {  F3(d) where d is Element of F2() : d in F1()  }  ) & ( x in  {  F3(d) where d is Element of F2() : d in F1()  }  implies x in  {  F3(d) where d is Element of D : d in F1()  }  ) )
hereby  ::_thesis: ( x in  {  F3(d) where d is Element of F2() : d in F1()  }  implies x in  {  F3(d) where d is Element of D : d in F1()  }  )
assume  x in  {  F3(d) where d is Element of D : d in F1()  }  ; ::_thesis: x in  {  F3(d) where d is Element of F2() : d in F1()  }
then  ex d being Element of D st
( F3(d) = x & d in F1() ) ;
hence  x in  {  F3(d) where d is Element of F2() : d in F1()  }  by A1; ::_thesis: verum
end;
hereby  ::_thesis: verum
assume  x in  {  F3(d) where d is Element of F2() : d in F1()  }  ; ::_thesis: x in  {  F3(d) where d is Element of D : d in F1()  }
then  ex d being Element of F2() st
( F3(d) = x & d in F1() ) ;
hence  x in  {  F3(d) where d is Element of D : d in F1()  }  ; ::_thesis: verum
end;
end;
A4: now__::_thesis:_for_d1,_d2_being_Element_of_D_st_F3(d1)_=_F3(d2)_holds_
d1_=_d2
let d1, d2 be Element of D; ::_thesis: ( F3(d1) = F3(d2) implies d1 = d2 )
 ( d1 in F1() & d2 in F1() ) ;
then reconsider d = d1, dd = d2 as Element of F2() by A1;
assume  F3(d1) = F3(d2) ; ::_thesis: d1 = d2
then  d = dd by A2;
hence  d1 = d2 ; ::_thesis: verum
end;
A5: F1() c= D ;
 F1(), {  F3(d) where d is Element of D : d in F1()  }  are_equipotent from FUNCT_7:sch_4(A5, A4);
hence  F1(), {  F3(d) where d is Element of F2() : d in F1()  }  are_equipotent by A3, TARSKI:1; ::_thesis: verum
end;
supposeA6: F1() = {} ; ::_thesis: F1(), {  F3(d) where d is Element of F2() : d in F1()  }  are_equipotent
now__::_thesis:_not__{__F3(d)_where_d_is_Element_of_F2()_:_d_in_F1()__}__<>_{}
set a = the Element of  {  F3(d) where d is Element of F2() : d in F1()  }  ;
assume   {  F3(d) where d is Element of F2() : d in F1()  }  <> {} ; ::_thesis: contradiction
then  the Element of  {  F3(d) where d is Element of F2() : d in F1()  }  in  {  F3(d) where d is Element of F2() : d in F1()  }  ;
then  ex d being Element of F2() st
( the Element of  {  F3(d) where d is Element of F2() : d in F1()  }  = F3(d) & d in F1() ) ;
hence  contradiction by A6; ::_thesis: verum
end;
hence  F1(), {  F3(d) where d is Element of F2() : d in F1()  }  are_equipotent by A6; ::_thesis: verum
end;
end;
end;

theorem :: FUNCT_7:104
for f, g, h being Function
for A being set st f,g equal_outside A holds
f +* h,g +* h equal_outside A
proof
let f, g, h be Function; ::_thesis: for A being set st f,g equal_outside A holds
f +* h,g +* h equal_outside A
let A be set ; ::_thesis: ( f,g equal_outside A implies f +* h,g +* h equal_outside A )
A1: dom ((f +* h) | ((dom (f +* h)) \ A)) = (dom (f +* h)) \ A by RELAT_1:156
.= ((dom f) \/ (dom h)) \ A by FUNCT_4:def_1
.= ((dom f) \ A) \/ ((dom h) \ A) by XBOOLE_1:42 ;
assume  f,g equal_outside A ; ::_thesis: f +* h,g +* h equal_outside A
then A2: f | ((dom f) \ A) = g | ((dom g) \ A) by Def2;
then A3: (dom f) \ A = dom (g | ((dom g) \ A)) by RELAT_1:156
.= (dom g) \ A by RELAT_1:156 ;
then A4: dom ((f +* h) | ((dom (f +* h)) \ A)) = ((dom g) \/ (dom h)) \ A by A1, XBOOLE_1:42
.= (dom (g +* h)) \ A by FUNCT_4:def_1
.= dom ((g +* h) | ((dom (g +* h)) \ A)) by RELAT_1:156 ;
now__::_thesis:_for_x_being_set_st_x_in_dom_((f_+*_h)_|_((dom_(f_+*_h))_\_A))_holds_
((f_+*_h)_|_((dom_(f_+*_h))_\_A))_._x_=_((g_+*_h)_|_((dom_(g_+*_h))_\_A))_._x
let x be set ; ::_thesis: ( x in dom ((f +* h) | ((dom (f +* h)) \ A)) implies ((f +* h) | ((dom (f +* h)) \ A)) . b1 = ((g +* h) | ((dom (g +* h)) \ A)) . b1 )
assume A5: x in dom ((f +* h) | ((dom (f +* h)) \ A)) ; ::_thesis: ((f +* h) | ((dom (f +* h)) \ A)) . b1 = ((g +* h) | ((dom (g +* h)) \ A)) . b1
then A6: ( x in (dom f) \ A or x in (dom h) \ A ) by A1, XBOOLE_0:def_3;
percases ( x in (dom h) \ A or not x in (dom h) \ A ) ;
supposeA7: x in (dom h) \ A ; ::_thesis: ((f +* h) | ((dom (f +* h)) \ A)) . b1 = ((g +* h) | ((dom (g +* h)) \ A)) . b1
thus ((f +* h) | ((dom (f +* h)) \ A)) . x = (f +* h) . x by A5, FUNCT_1:47
.= h . x by A7, FUNCT_4:13
.= (g +* h) . x by A7, FUNCT_4:13
.= ((g +* h) | ((dom (g +* h)) \ A)) . x by A4, A5, FUNCT_1:47 ; ::_thesis: verum
end;
supposeA8: not x in (dom h) \ A ; ::_thesis: ((f +* h) | ((dom (f +* h)) \ A)) . b1 = ((g +* h) | ((dom (g +* h)) \ A)) . b1
 not x in A by A6, XBOOLE_0:def_5;
then A9: not x in dom h by A8, XBOOLE_0:def_5;
A10: x in (dom f) \ A by A1, A5, A8, XBOOLE_0:def_3;
thus ((f +* h) | ((dom (f +* h)) \ A)) . x = (f +* h) . x by A5, FUNCT_1:47
.= f . x by A9, FUNCT_4:11
.= (g | ((dom g) \ A)) . x by A2, A6, A8, FUNCT_1:49
.= g . x by A3, A10, FUNCT_1:49
.= (g +* h) . x by A9, FUNCT_4:11
.= ((g +* h) | ((dom (g +* h)) \ A)) . x by A4, A5, FUNCT_1:47 ; ::_thesis: verum
end;
end;
end;
then  (f +* h) | ((dom (f +* h)) \ A) = (g +* h) | ((dom (g +* h)) \ A) by A4, FUNCT_1:2;
hence  f +* h,g +* h equal_outside A by Def2; ::_thesis: verum
end;

theorem :: FUNCT_7:105
for f, g, h being Function
for A being set st f,g equal_outside A holds
h +* f,h +* g equal_outside A
proof
let f, g, h be Function; ::_thesis: for A being set st f,g equal_outside A holds
h +* f,h +* g equal_outside A
let A be set ; ::_thesis: ( f,g equal_outside A implies h +* f,h +* g equal_outside A )
assume  f,g equal_outside A ; ::_thesis: h +* f,h +* g equal_outside A
then A1: f | ((dom f) \ A) = g | ((dom g) \ A) by Def2;
then A2: (dom f) \ A = dom (g | ((dom g) \ A)) by RELAT_1:156
.= (dom g) \ A by RELAT_1:156 ;
A3: dom ((h +* f) | ((dom (h +* f)) \ A)) = (dom (h +* f)) \ A by RELAT_1:156
.= ((dom h) \/ (dom f)) \ A by FUNCT_4:def_1
.= ((dom h) \ A) \/ ((dom f) \ A) by XBOOLE_1:42 ;
then A4: dom ((h +* f) | ((dom (h +* f)) \ A)) = ((dom h) \/ (dom g)) \ A by A2, XBOOLE_1:42
.= (dom (h +* g)) \ A by FUNCT_4:def_1
.= dom ((h +* g) | ((dom (h +* g)) \ A)) by RELAT_1:156 ;
now__::_thesis:_for_x_being_set_st_x_in_dom_((h_+*_f)_|_((dom_(h_+*_f))_\_A))_holds_
((h_+*_f)_|_((dom_(h_+*_f))_\_A))_._x_=_((h_+*_g)_|_((dom_(h_+*_g))_\_A))_._x
let x be set ; ::_thesis: ( x in dom ((h +* f) | ((dom (h +* f)) \ A)) implies ((h +* f) | ((dom (h +* f)) \ A)) . b1 = ((h +* g) | ((dom (h +* g)) \ A)) . b1 )
assume A5: x in dom ((h +* f) | ((dom (h +* f)) \ A)) ; ::_thesis: ((h +* f) | ((dom (h +* f)) \ A)) . b1 = ((h +* g) | ((dom (h +* g)) \ A)) . b1
then A6: ( x in (dom h) \ A or x in (dom f) \ A ) by A3, XBOOLE_0:def_3;
percases ( x in (dom f) \ A or not x in (dom f) \ A ) ;
supposeA7: x in (dom f) \ A ; ::_thesis: ((h +* f) | ((dom (h +* f)) \ A)) . b1 = ((h +* g) | ((dom (h +* g)) \ A)) . b1
thus ((h +* f) | ((dom (h +* f)) \ A)) . x = (h +* f) . x by A5, FUNCT_1:47
.= f . x by A7, FUNCT_4:13
.= (g | ((dom g) \ A)) . x by A1, A7, FUNCT_1:49
.= g . x by A2, A7, FUNCT_1:49
.= (h +* g) . x by A2, A7, FUNCT_4:13
.= ((h +* g) | ((dom (h +* g)) \ A)) . x by A4, A5, FUNCT_1:47 ; ::_thesis: verum
end;
supposeA8: not x in (dom f) \ A ; ::_thesis: ((h +* f) | ((dom (h +* f)) \ A)) . b1 = ((h +* g) | ((dom (h +* g)) \ A)) . b1
A9: not x in A by A6, XBOOLE_0:def_5;
then A10: not x in dom f by A8, XBOOLE_0:def_5;
A11: not x in dom g by A2, A8, A9, XBOOLE_0:def_5;
thus ((h +* f) | ((dom (h +* f)) \ A)) . x = (h +* f) . x by A5, FUNCT_1:47
.= h . x by A10, FUNCT_4:11
.= (h +* g) . x by A11, FUNCT_4:11
.= ((h +* g) | ((dom (h +* g)) \ A)) . x by A4, A5, FUNCT_1:47 ; ::_thesis: verum
end;
end;
end;
then  (h +* f) | ((dom (h +* f)) \ A) = (h +* g) | ((dom (h +* g)) \ A) by A4, FUNCT_1:2;
hence  h +* f,h +* g equal_outside A by Def2; ::_thesis: verum
end;

theorem :: FUNCT_7:106
for f, g, h being Function holds
( f +* h = g +* h iff f,g equal_outside dom h )
proof
let f, g, h be Function; ::_thesis: ( f +* h = g +* h iff f,g equal_outside dom h )
thus  ( f +* h = g +* h implies f,g equal_outside dom h ) ::_thesis: ( f,g equal_outside dom h implies f +* h = g +* h )
proof
assume  f +* h = g +* h ; ::_thesis: f,g equal_outside dom h
then A1: f +* h,g equal_outside dom h by Th26, Th29;
 f,f +* h equal_outside dom h by Th29;
hence  f,g equal_outside dom h by A1, Th27; ::_thesis: verum
end;
assume A2: f,g equal_outside dom h ; ::_thesis: f +* h = g +* h
then A3: (dom f) \ (dom h) = (dom g) \ (dom h) by Th28;
A4: for x being set st x in dom (f +* h) holds
(f +* h) . x = (g +* h) . x
proof
let x be set ; ::_thesis: ( x in dom (f +* h) implies (f +* h) . x = (g +* h) . x )
assume A5: x in dom (f +* h) ; ::_thesis: (f +* h) . x = (g +* h) . x
percases ( x in dom h or not x in dom h ) ;
supposeA6: x in dom h ; ::_thesis: (f +* h) . x = (g +* h) . x
hence (f +* h) . x = h . x by FUNCT_4:13
.= (g +* h) . x by A6, FUNCT_4:13 ;
::_thesis: verum
end;
supposeA7: not x in dom h ; ::_thesis: (f +* h) . x = (g +* h) . x
 dom (f +* h) = (dom f) \/ (dom h) by FUNCT_4:def_1;
then  x in dom f by A5, A7, XBOOLE_0:def_3;
then A8: x in (dom f) \ (dom h) by A7, XBOOLE_0:def_5;
A9: f | ((dom f) \ (dom h)) = g | ((dom g) \ (dom h)) by A2, Def2;
thus (f +* h) . x = f . x by A7, FUNCT_4:11
.= (g | ((dom g) \ (dom h))) . x by A9, A8, FUNCT_1:49
.= g . x by A3, A8, FUNCT_1:49
.= (g +* h) . x by A7, FUNCT_4:11 ; ::_thesis: verum
end;
end;
end;
dom (f +* h) = (dom f) \/ (dom h) by FUNCT_4:def_1
.= ((dom g) \ (dom h)) \/ (dom h) by A3, XBOOLE_1:39
.= (dom g) \/ (dom h) by XBOOLE_1:39
.= dom (g +* h) by FUNCT_4:def_1 ;
hence  f +* h = g +* h by A4, FUNCT_1:2; ::_thesis: verum
end;

theorem Th107: :: FUNCT_7:107
for x, y, a being set
for f being Function st f . x = f . y holds
f . a = (f * ((id (dom f)) +* (x,y))) . a
proof
let x, y, a be set ; ::_thesis: for f being Function st f . x = f . y holds
f . a = (f * ((id (dom f)) +* (x,y))) . a
let f be Function; ::_thesis: ( f . x = f . y implies f . a = (f * ((id (dom f)) +* (x,y))) . a )
assume A1: f . x = f . y ; ::_thesis: f . a = (f * ((id (dom f)) +* (x,y))) . a
set g1 = (id (dom f)) +* (x,y);
A2: dom (id (dom f)) = dom f ;
percases ( not x in dom f or x in dom f ) ;
suppose not x in dom f ; ::_thesis: f . a = (f * ((id (dom f)) +* (x,y))) . a
then  id (dom f) = (id (dom f)) +* (x,y) by A2, Def3;
hence  f . a = (f * ((id (dom f)) +* (x,y))) . a by RELAT_1:52; ::_thesis: verum
end;
supposeA3: x in dom f ; ::_thesis: f . a = (f * ((id (dom f)) +* (x,y))) . a
A4: dom ((id (dom f)) +* (x,y)) = dom f by A2, Th30;
A5: ((id (dom f)) +* (x,y)) . x = y by A2, A3, Th31;
thus  f . a = (f * ((id (dom f)) +* (x,y))) . a ::_thesis: verum
proof
percases ( a in dom f or not a in dom f ) ;
supposeA6: a in dom f ; ::_thesis: f . a = (f * ((id (dom f)) +* (x,y))) . a
now__::_thesis:_(_a_<>_x_implies_f_._a_=_(f_*_((id_(dom_f))_+*_(x,y)))_._a_)
assume  a <> x ; ::_thesis: f . a = (f * ((id (dom f)) +* (x,y))) . a
then ((id (dom f)) +* (x,y)) . a = (id (dom f)) . a by Th32
.= a by A6, FUNCT_1:18 ;
hence  f . a = (f * ((id (dom f)) +* (x,y))) . a by A4, A6, FUNCT_1:13; ::_thesis: verum
end;
hence  f . a = (f * ((id (dom f)) +* (x,y))) . a by A1, A3, A5, A4, FUNCT_1:13; ::_thesis: verum
end;
supposeA7: not a in dom f ; ::_thesis: f . a = (f * ((id (dom f)) +* (x,y))) . a
 dom (f * ((id (dom f)) +* (x,y))) c= dom ((id (dom f)) +* (x,y)) by RELAT_1:25;
then  not a in dom (f * ((id (dom f)) +* (x,y))) by A4, A7;
then  (f * ((id (dom f)) +* (x,y))) . a = {} by FUNCT_1:def_2;
hence  f . a = (f * ((id (dom f)) +* (x,y))) . a by A7, FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;
end;
end;

theorem :: FUNCT_7:108
for x, y being set
for f being Function st ( x in dom f implies ( y in dom f & f . x = f . y ) ) holds
f = f * ((id (dom f)) +* (x,y))
proof
let x, y be set ; ::_thesis: for f being Function st ( x in dom f implies ( y in dom f & f . x = f . y ) ) holds
f = f * ((id (dom f)) +* (x,y))
let f be Function; ::_thesis: ( ( x in dom f implies ( y in dom f & f . x = f . y ) ) implies f = f * ((id (dom f)) +* (x,y)) )
assume A1: ( x in dom f implies ( y in dom f & f . x = f . y ) ) ; ::_thesis: f = f * ((id (dom f)) +* (x,y))
set g1 = (id (dom f)) +* (x,y);
set g = f * ((id (dom f)) +* (x,y));
A2: dom (id (dom f)) = dom f ;
percases ( not x in dom f or x in dom f ) ;
suppose not x in dom f ; ::_thesis: f = f * ((id (dom f)) +* (x,y))
then  id (dom f) = (id (dom f)) +* (x,y) by A2, Def3;
hence  f = f * ((id (dom f)) +* (x,y)) by RELAT_1:52; ::_thesis: verum
end;
supposeA3: x in dom f ; ::_thesis: f = f * ((id (dom f)) +* (x,y))
A4: dom ((id (dom f)) +* (x,y)) = dom f by A2, Th30;
now__::_thesis:_(_dom_f_=_dom_(f_*_((id_(dom_f))_+*_(x,y)))_&_(_for_a_being_set_st_a_in_dom_f_holds_
f_._a_=_(f_*_((id_(dom_f))_+*_(x,y)))_._a_)_)
 rng ((id (dom f)) +* (x,y)) c= dom f
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng ((id (dom f)) +* (x,y)) or b in dom f )
assume  b in rng ((id (dom f)) +* (x,y)) ; ::_thesis: b in dom f
then consider a being set  such that
A5: a in dom ((id (dom f)) +* (x,y)) and
A6: b = ((id (dom f)) +* (x,y)) . a by FUNCT_1:def_3;
percases ( a = x or a <> x ) ;
suppose a = x ; ::_thesis: b in dom f
hence  b in dom f by A1, A2, A3, A6, Th31; ::_thesis: verum
end;
suppose a <> x ; ::_thesis: b in dom f
then  (id (dom f)) . a = ((id (dom f)) +* (x,y)) . a by Th32;
hence  b in dom f by A4, A5, A6, FUNCT_1:18; ::_thesis: verum
end;
end;
end;
hence  dom f = dom (f * ((id (dom f)) +* (x,y))) by A4, RELAT_1:27; ::_thesis: for a being set st a in dom f holds
f . a = (f * ((id (dom f)) +* (x,y))) . a
let a be set ; ::_thesis: ( a in dom f implies f . a = (f * ((id (dom f)) +* (x,y))) . a )
assume  a in dom f ; ::_thesis: f . a = (f * ((id (dom f)) +* (x,y))) . a
thus  f . a = (f * ((id (dom f)) +* (x,y))) . a by A1, A3, Th107; ::_thesis: verum
end;
hence  f = f * ((id (dom f)) +* (x,y)) by FUNCT_1:2; ::_thesis: verum
end;
end;
end;

theorem :: FUNCT_7:109
for f being Function
for x being set st x in dom f holds
f +* (x .--> (f . x)) = f
proof
let f be Function; ::_thesis: for x being set st x in dom f holds
f +* (x .--> (f . x)) = f
let x be set ; ::_thesis: ( x in dom f implies f +* (x .--> (f . x)) = f )
assume  x in dom f ; ::_thesis: f +* (x .--> (f . x)) = f
hence f +* (x .--> (f . x)) = f +* (x,(f . x)) by Def3
.= f by Th35 ;
::_thesis: verum
end;

theorem Th110: :: FUNCT_7:110
for X being set
for p being Permutation of X
for x, y being Element of X holds (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X
proof
let X be set ; ::_thesis: for p being Permutation of X
for x, y being Element of X holds (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X
let p be Permutation of X; ::_thesis: for x, y being Element of X holds (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X
let x, y be Element of X; ::_thesis: (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X
set p1 = p +* (x,(p . y));
set p2 = (p +* (x,(p . y))) +* (y,(p . x));
A1: dom ((p +* (x,(p . y))) +* (y,(p . x))) = dom (p +* (x,(p . y))) by Th30;
A2: dom (p +* (x,(p . y))) = dom p by Th30;
A3: ( X = {} implies X = {} ) ;
then A4: dom p = X by FUNCT_2:def_1;
percases ( X is empty or not X is empty ) ;
suppose X is empty ; ::_thesis: (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X
then  (p +* (x,(p . y))) +* (y,(p . x)) = {} by A1, A2;
hence  (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X by A1, A2, A3; ::_thesis: verum
end;
supposeA5: not X is empty ; ::_thesis: (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X
A6: rng p = X by FUNCT_2:def_3;
then A7: p . x in X by A4, A5, FUNCT_1:def_3;
thus  (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X ::_thesis: verum
proof
percases ( x = y or x <> y ) ;
supposeA8: x = y ; ::_thesis: (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X
then (p +* (x,(p . y))) +* (y,(p . x)) = p +* (x,(p . y)) by Th35
.= p by A8, Th35 ;
hence  (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X ; ::_thesis: verum
end;
supposeA9: x <> y ; ::_thesis: (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X
then A10: ((p +* (x,(p . y))) +* (y,(p . x))) . x = (p +* (x,(p . y))) . x by Th32
.= p . y by A4, A5, Th31 ;
A11: now__::_thesis:_for_z_being_set_st_z_in_X_&_z_<>_x_&_z_<>_y_holds_
((p_+*_(x,(p_._y)))_+*_(y,(p_._x)))_._z_=_p_._z
let z be set ; ::_thesis: ( z in X & z <> x & z <> y implies ((p +* (x,(p . y))) +* (y,(p . x))) . z = p . z )
assume that
 z in X and
A12: z <> x and
A13: z <> y ; ::_thesis: ((p +* (x,(p . y))) +* (y,(p . x))) . z = p . z
thus ((p +* (x,(p . y))) +* (y,(p . x))) . z = (p +* (x,(p . y))) . z by A13, Th32
.= p . z by A12, Th32 ; ::_thesis: verum
end;
A14: ((p +* (x,(p . y))) +* (y,(p . x))) . y = p . x by A2, A4, A5, Th31;
A15: now__::_thesis:_for_pz_being_set_holds_
(_(_pz_in_rng_((p_+*_(x,(p_._y)))_+*_(y,(p_._x)))_implies_pz_in_X_)_&_(_pz_in_X_implies_pz_in_rng_((p_+*_(x,(p_._y)))_+*_(y,(p_._x)))_)_)
let pz be set ; ::_thesis: ( ( pz in rng ((p +* (x,(p . y))) +* (y,(p . x))) implies pz in X ) & ( pz in X implies b1 in rng ((p +* (x,(p . y))) +* (y,(p . x))) ) )
hereby  ::_thesis: ( pz in X implies b1 in rng ((p +* (x,(p . y))) +* (y,(p . x))) )
assume  pz in rng ((p +* (x,(p . y))) +* (y,(p . x))) ; ::_thesis: pz in X
then consider z being set  such that
A16: z in dom ((p +* (x,(p . y))) +* (y,(p . x))) and
A17: pz = ((p +* (x,(p . y))) +* (y,(p . x))) . z by FUNCT_1:def_3;
A18: p . z in X by A1, A2, A6, A16, FUNCT_1:def_3;
percases ( z = x or z = y or ( z <> x & z <> y ) ) ;
suppose z = x ; ::_thesis: pz in X
hence  pz in X by A4, A5, A6, A10, A17, FUNCT_1:def_3; ::_thesis: verum
end;
suppose z = y ; ::_thesis: pz in X
hence  pz in X by A2, A4, A7, A17, Th31; ::_thesis: verum
end;
suppose ( z <> x & z <> y ) ; ::_thesis: pz in X
hence  pz in X by A1, A2, A11, A16, A17, A18; ::_thesis: verum
end;
end;
end;
assume  pz in X ; ::_thesis: b1 in rng ((p +* (x,(p . y))) +* (y,(p . x)))
then consider z being set  such that
A19: z in dom p and
A20: pz = p . z by A6, FUNCT_1:def_3;
percases ( z = x or z = y or ( z <> x & z <> y ) ) ;
suppose z = x ; ::_thesis: b1 in rng ((p +* (x,(p . y))) +* (y,(p . x)))
hence  pz in rng ((p +* (x,(p . y))) +* (y,(p . x))) by A1, A2, A4, A14, A19, A20, FUNCT_1:def_3; ::_thesis: verum
end;
suppose z = y ; ::_thesis: b1 in rng ((p +* (x,(p . y))) +* (y,(p . x)))
hence  pz in rng ((p +* (x,(p . y))) +* (y,(p . x))) by A1, A2, A4, A10, A19, A20, FUNCT_1:def_3; ::_thesis: verum
end;
suppose ( z <> x & z <> y ) ; ::_thesis: b1 in rng ((p +* (x,(p . y))) +* (y,(p . x)))
then  ((p +* (x,(p . y))) +* (y,(p . x))) . z = p . z by A11, A19;
hence  pz in rng ((p +* (x,(p . y))) +* (y,(p . x))) by A1, A2, A19, A20, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
then  rng ((p +* (x,(p . y))) +* (y,(p . x))) = X by TARSKI:1;
then reconsider p2 = (p +* (x,(p . y))) +* (y,(p . x)) as Function of X,X by A1, A2, A4, A5, FUNCT_2:def_1, RELSET_1:4;
now__::_thesis:_for_z1,_z2_being_set_st_z1_in_X_&_z2_in_X_&_p2_._z1_=_p2_._z2_holds_
not_z1_<>_z2
let z1, z2 be set ; ::_thesis: ( z1 in X & z2 in X & p2 . z1 = p2 . z2 implies not z1 <> z2 )
assume that
A21: z1 in X and
A22: z2 in X and
A23: p2 . z1 = p2 . z2 and
A24: z1 <> z2 ; ::_thesis: contradiction
percases ( ( z1 = x & z2 = y ) or ( z1 = y & z2 = x ) or ( z1 = x & z2 <> y ) or ( z1 <> x & z2 = y ) or ( z1 = y & z2 <> x ) or ( z1 <> y & z2 = x ) or ( z1 <> y & z2 <> x & z1 <> x & z2 <> y ) ) ;
suppose ( z1 = x & z2 = y ) ; ::_thesis: contradiction
hence  contradiction by A5, A9, A10, A14, A23, FUNCT_2:19; ::_thesis: verum
end;
suppose ( z1 = y & z2 = x ) ; ::_thesis: contradiction
hence  contradiction by A5, A9, A10, A14, A23, FUNCT_2:19; ::_thesis: verum
end;
supposeA25: ( z1 = x & z2 <> y ) ; ::_thesis: contradiction
then  p2 . z2 = p . z2 by A11, A22, A24;
hence  contradiction by A10, A22, A23, A25, FUNCT_2:19; ::_thesis: verum
end;
supposeA26: ( z1 <> x & z2 = y ) ; ::_thesis: contradiction
then  p2 . z1 = p . z1 by A11, A21, A24;
hence  contradiction by A14, A21, A23, A26, FUNCT_2:19; ::_thesis: verum
end;
supposeA27: ( z1 = y & z2 <> x ) ; ::_thesis: contradiction
then  p2 . z2 = p . z2 by A11, A22, A24;
hence  contradiction by A14, A22, A23, A27, FUNCT_2:19; ::_thesis: verum
end;
supposeA28: ( z1 <> y & z2 = x ) ; ::_thesis: contradiction
then  p2 . z1 = p . z1 by A11, A21, A24;
hence  contradiction by A10, A21, A23, A28, FUNCT_2:19; ::_thesis: verum
end;
suppose ( z1 <> y & z2 <> x & z1 <> x & z2 <> y ) ; ::_thesis: contradiction
then  ( p2 . z1 = p . z1 & p2 . z2 = p . z2 ) by A11, A21, A22;
hence  contradiction by A21, A22, A23, A24, FUNCT_2:19; ::_thesis: verum
end;
end;
end;
then A29: p2 is one-to-one by A5, FUNCT_2:19;
 rng p2 = X by A15, TARSKI:1;
then  p2 is onto by FUNCT_2:def_3;
hence  (p +* (x,(p . y))) +* (y,(p . x)) is Permutation of X by A29; ::_thesis: verum
end;
end;
end;
end;
end;
end;

theorem :: FUNCT_7:111
for f being Function
for x, y being set st x in dom f & y in dom f holds
ex p being Permutation of (dom f) st (f +* (x,(f . y))) +* (y,(f . x)) = f * p
proof
let f be Function; ::_thesis: for x, y being set st x in dom f & y in dom f holds
ex p being Permutation of (dom f) st (f +* (x,(f . y))) +* (y,(f . x)) = f * p
let x, y be set ; ::_thesis: ( x in dom f & y in dom f implies ex p being Permutation of (dom f) st (f +* (x,(f . y))) +* (y,(f . x)) = f * p )
assume that
A1: x in dom f and
A2: y in dom f ; ::_thesis: ex p being Permutation of (dom f) st (f +* (x,(f . y))) +* (y,(f . x)) = f * p
set i = id (dom f);
 ( (id (dom f)) . x = x & (id (dom f)) . y = y ) by A1, A2, FUNCT_1:18;
then reconsider p = ((id (dom f)) +* (x,y)) +* (y,x) as Permutation of (dom f) by A1, A2, Th110;
set pk = (id (dom f)) +* (x,y);
set fr = f * p;
set fk = f +* (x,(f . y));
take  p ; ::_thesis: (f +* (x,(f . y))) +* (y,(f . x)) = f * p
set fl = (f +* (x,(f . y))) +* (y,(f . x));
A3: dom (id (dom f)) = dom f ;
A4: dom (f +* (x,(f . y))) = dom ((f +* (x,(f . y))) +* (y,(f . x))) by Th30;
A5: dom p = dom ((id (dom f)) +* (x,y)) by Th30;
A6: dom ((id (dom f)) +* (x,y)) = dom (id (dom f)) by Th30;
A7: dom f = dom (f +* (x,(f . y))) by Th30;
now__::_thesis:_(_dom_f_=_dom_((f_+*_(x,(f_._y)))_+*_(y,(f_._x)))_&_dom_f_=_dom_(f_*_p)_&_(_for_z_being_set_st_z_in_dom_f_holds_
((f_+*_(x,(f_._y)))_+*_(y,(f_._x)))_._z_=_(f_*_p)_._z_)_)
thus  dom f = dom ((f +* (x,(f . y))) +* (y,(f . x))) by A4, Th30; ::_thesis: ( dom f = dom (f * p) & ( for z being set st z in dom f holds
((f +* (x,(f . y))) +* (y,(f . x))) . b2 = (f * p) . b2 ) )
 rng p = dom f by FUNCT_2:def_3;
hence  dom f = dom (f * p) by A5, A6, RELAT_1:27; ::_thesis: for z being set st z in dom f holds
((f +* (x,(f . y))) +* (y,(f . x))) . b2 = (f * p) . b2
let z be set ; ::_thesis: ( z in dom f implies ((f +* (x,(f . y))) +* (y,(f . x))) . b1 = (f * p) . b1 )
assume A8: z in dom f ; ::_thesis: ((f +* (x,(f . y))) +* (y,(f . x))) . b1 = (f * p) . b1
percases ( x <> y or x = y ) ;
supposeA9: x <> y ; ::_thesis: ((f +* (x,(f . y))) +* (y,(f . x))) . b1 = (f * p) . b1
thus  ((f +* (x,(f . y))) +* (y,(f . x))) . z = (f * p) . z ::_thesis: verum
proof
percases ( z = x or z = y or ( z <> x & z <> y ) ) ;
supposeA10: z = x ; ::_thesis: ((f +* (x,(f . y))) +* (y,(f . x))) . z = (f * p) . z
hence ((f +* (x,(f . y))) +* (y,(f . x))) . z = (f +* (x,(f . y))) . z by A9, Th32
.= f . y by A8, A10, Th31
.= f . (((id (dom f)) +* (x,y)) . x) by A1, A3, Th31
.= f . (p . x) by A9, Th32
.= (f * p) . z by A5, A6, A8, A10, FUNCT_1:13 ;
::_thesis: verum
end;
supposeA11: z = y ; ::_thesis: ((f +* (x,(f . y))) +* (y,(f . x))) . z = (f * p) . z
hence ((f +* (x,(f . y))) +* (y,(f . x))) . z = f . x by A7, A8, Th31
.= f . (p . y) by A2, A6, Th31
.= (f * p) . z by A5, A6, A8, A11, FUNCT_1:13 ;
::_thesis: verum
end;
supposeA12: ( z <> x & z <> y ) ; ::_thesis: ((f +* (x,(f . y))) +* (y,(f . x))) . z = (f * p) . z
then A13: p . z = ((id (dom f)) +* (x,y)) . z by Th32
.= (id (dom f)) . z by A12, Th32
.= z by A8, FUNCT_1:18 ;
thus ((f +* (x,(f . y))) +* (y,(f . x))) . z = (f +* (x,(f . y))) . z by A12, Th32
.= f . (p . z) by A12, A13, Th32
.= (f * p) . z by A5, A6, A8, FUNCT_1:13 ; ::_thesis: verum
end;
end;
end;
end;
supposeA14: x = y ; ::_thesis: ((f +* (x,(f . y))) +* (y,(f . x))) . b1 = (f * p) . b1
A15: x = (id (dom f)) . x by A1, FUNCT_1:17;
 ( f +* (x,(f . y)) = f & id (dom f) = (id (dom f)) +* (x,((id (dom f)) . y)) ) by A14, Th35;
hence  ((f +* (x,(f . y))) +* (y,(f . x))) . z = (f * p) . z by A14, A15, RELAT_1:52; ::_thesis: verum
end;
end;
end;
hence  (f +* (x,(f . y))) +* (y,(f . x)) = f * p by FUNCT_1:2; ::_thesis: verum
end;

theorem :: FUNCT_7:112
for f being Function
for d, r being set st d in dom f holds
dom f = dom (f +* (d .--> r))
proof
let f be Function; ::_thesis: for d, r being set st d in dom f holds
dom f = dom (f +* (d .--> r))
let d, r be set ; ::_thesis: ( d in dom f implies dom f = dom (f +* (d .--> r)) )
assume A1: d in dom f ; ::_thesis: dom f = dom (f +* (d .--> r))
thus  dom f = dom (f +* (d,r)) by Th30
.= dom (f +* (d .--> r)) by A1, Def3 ; ::_thesis: verum
end;

theorem :: FUNCT_7:113
for f, g being FinSequence of INT
for m, n being Element of NAT st 1 <= n & n <= len f & 1 <= m & m <= len f & g = (f +* (m,(f /. n))) +* (n,(f /. m)) holds
( f . m = g . n & f . n = g . m & ( for k being set st k <> m & k <> n & k in dom f holds
f . k = g . k ) & f,g are_fiberwise_equipotent )
proof
let f, g be FinSequence of INT ; ::_thesis: for m, n being Element of NAT st 1 <= n & n <= len f & 1 <= m & m <= len f & g = (f +* (m,(f /. n))) +* (n,(f /. m)) holds
( f . m = g . n & f . n = g . m & ( for k being set st k <> m & k <> n & k in dom f holds
f . k = g . k ) & f,g are_fiberwise_equipotent )
let m, n be Element of NAT ; ::_thesis: ( 1 <= n & n <= len f & 1 <= m & m <= len f & g = (f +* (m,(f /. n))) +* (n,(f /. m)) implies ( f . m = g . n & f . n = g . m & ( for k being set st k <> m & k <> n & k in dom f holds
f . k = g . k ) & f,g are_fiberwise_equipotent ) )
assume that
A1: ( 1 <= n & n <= len f ) and
A2: ( 1 <= m & m <= len f ) and
A3: g = (f +* (m,(f /. n))) +* (n,(f /. m)) ; ::_thesis: ( f . m = g . n & f . n = g . m & ( for k being set st k <> m & k <> n & k in dom f holds
f . k = g . k ) & f,g are_fiberwise_equipotent )
A4: dom (f +* (m,(f /. n))) = dom f by Th30;
A5: n in dom f by A1, FINSEQ_3:25;
hence A6: g . n = f /. m by A3, A4, Th31
.= f . m by A2, FINSEQ_4:15 ;
::_thesis: ( f . n = g . m & ( for k being set st k <> m & k <> n & k in dom f holds
f . k = g . k ) & f,g are_fiberwise_equipotent )
A7: m in dom f by A2, FINSEQ_3:25;
hereby  ::_thesis: ( ( for k being set st k <> m & k <> n & k in dom f holds
f . k = g . k ) & f,g are_fiberwise_equipotent )
percases ( m = n or m <> n ) ;
suppose m = n ; ::_thesis: g . m = f . n
hence  g . m = f . n by A6; ::_thesis: verum
end;
suppose m <> n ; ::_thesis: g . m = f . n
hence g . m = (f +* (m,(f /. n))) . m by A3, Th32
.= f /. n by A7, Th31
.= f . n by A1, FINSEQ_4:15 ;
::_thesis: verum
end;
end;
end;
A9: now__::_thesis:_for_k_being_set_st_k_<>_m_&_k_<>_n_&_k_in_dom_f_holds_
g_._k_=_f_._k
let k be set ; ::_thesis: ( k <> m & k <> n & k in dom f implies g . k = f . k )
assume that
A10: k <> m and
A11: k <> n and
 k in dom f ; ::_thesis: g . k = f . k
thus g . k = (f +* (m,(f /. n))) . k by A3, A11, Th32
.= f . k by A10, Th32 ; ::_thesis: verum
end;
 dom g = dom f by A3, A4, Th30;
hence  ( ( for k being set st k <> m & k <> n & k in dom f holds
f . k = g . k ) & f,g are_fiberwise_equipotent ) by A5, A7, A6, A8, A9, RFINSEQ:28; ::_thesis: verum
end;

theorem :: FUNCT_7:114
for f being Function
for a, A, b, B, c, C being set st a <> b & a <> c holds
(((f +* (a .--> A)) +* (b .--> B)) +* (c .--> C)) . a = A
proof
let f be Function; ::_thesis: for a, A, b, B, c, C being set st a <> b & a <> c holds
(((f +* (a .--> A)) +* (b .--> B)) +* (c .--> C)) . a = A
let a, A, b, B, c, C be set ; ::_thesis: ( a <> b & a <> c implies (((f +* (a .--> A)) +* (b .--> B)) +* (c .--> C)) . a = A )
assume  ( a <> b & a <> c ) ; ::_thesis: (((f +* (a .--> A)) +* (b .--> B)) +* (c .--> C)) . a = A
hence (((f +* (a .--> A)) +* (b .--> B)) +* (c .--> C)) . a = (f +* (a .--> A)) . a by FUNCT_4:91
.= A by Th94 ;
::_thesis: verum
end;

theorem Th115: :: FUNCT_7:115
for A, B, a, b being set
for f being Function of A,B st b in B holds
f +* (a,b) is Function of A,B
proof
let A, B, a, b be set ; ::_thesis: for f being Function of A,B st b in B holds
f +* (a,b) is Function of A,B
let f be Function of A,B; ::_thesis: ( b in B implies f +* (a,b) is Function of A,B )
assume A1: b in B ; ::_thesis: f +* (a,b) is Function of A,B
A2: dom f = A by A1, FUNCT_2:def_1;
percases ( not a in A or a in A ) ;
suppose not a in A ; ::_thesis: f +* (a,b) is Function of A,B
hence  f +* (a,b) is Function of A,B by A2, Def3; ::_thesis: verum
end;
supposeA3: a in A ; ::_thesis: f +* (a,b) is Function of A,B
set g = a .--> b;
set f1 = f +* (a .--> b);
 rng (a .--> b) = {b} by FUNCOP_1:8;
then  rng (a .--> b) c= B by A1, ZFMISC_1:31;
then A4: ( rng (f +* (a .--> b)) c= (rng f) \/ (rng (a .--> b)) & (rng f) \/ (rng (a .--> b)) c= B \/ B ) by FUNCT_4:17, XBOOLE_1:13;
A5: {a} c= A by A3, ZFMISC_1:31;
dom (f +* (a .--> b)) = A \/ (dom (a .--> b)) by A2, FUNCT_4:def_1
.= A \/ {a} by FUNCOP_1:13
.= A by A5, XBOOLE_1:12 ;
then A6: f +* (a .--> b) is Function of A,B by A4, FUNCT_2:2, XBOOLE_1:1;
 a in dom f by A1, A3, FUNCT_2:def_1;
hence  f +* (a,b) is Function of A,B by A6, Def3; ::_thesis: verum
end;
end;
end;

theorem Th116: :: FUNCT_7:116
for A being set
for f being Function
for x, y being set st rng f c= A holds
f +~ (x,y) = ((id A) +* (x,y)) * f
proof
let A be set ; ::_thesis: for f being Function
for x, y being set st rng f c= A holds
f +~ (x,y) = ((id A) +* (x,y)) * f
let f be Function; ::_thesis: for x, y being set st rng f c= A holds
f +~ (x,y) = ((id A) +* (x,y)) * f
let x, y be set ; ::_thesis: ( rng f c= A implies f +~ (x,y) = ((id A) +* (x,y)) * f )
assume A1: rng f c= A ; ::_thesis: f +~ (x,y) = ((id A) +* (x,y)) * f
percases ( x in A or not x in A ) ;
suppose x in A ; ::_thesis: f +~ (x,y) = ((id A) +* (x,y)) * f
then A2: x in dom (id A) ;
thus f +~ (x,y) = ((id A) * f) +* ((x .--> y) * f) by A1, RELAT_1:53
.= ((id A) +* (x .--> y)) * f by Th10
.= ((id A) +* (x,y)) * f by A2, Def3 ; ::_thesis: verum
end;
supposeA3: not x in A ; ::_thesis: f +~ (x,y) = ((id A) +* (x,y)) * f
then A4: not x in dom (id A) ;
 not x in rng f by A1, A3;
hence f +~ (x,y) = f by FUNCT_4:103
.= (id A) * f by A1, RELAT_1:53
.= ((id A) +* (x,y)) * f by A4, Def3 ;
::_thesis: verum
end;
end;
end;

theorem :: FUNCT_7:117
for f, g being Function
for x, y being set holds (f +* g) +~ (x,y) = (f +~ (x,y)) +* (g +~ (x,y))
proof
let f, g be Function; ::_thesis: for x, y being set holds (f +* g) +~ (x,y) = (f +~ (x,y)) +* (g +~ (x,y))
let x, y be set ; ::_thesis: (f +* g) +~ (x,y) = (f +~ (x,y)) +* (g +~ (x,y))
set A = (rng f) \/ (rng g);
A1: g +~ (x,y) = ((id ((rng f) \/ (rng g))) +* (x,y)) * g by Th116, XBOOLE_1:7;
A2: dom ((id ((rng f) \/ (rng g))) +* (x,y)) = dom (id ((rng f) \/ (rng g))) by Th30
.= (rng f) \/ (rng g) ;
 rng (f +* g) c= (rng f) \/ (rng g) by FUNCT_4:17;
hence (f +* g) +~ (x,y) = ((id ((rng f) \/ (rng g))) +* (x,y)) * (f +* g) by Th116
.= (((id ((rng f) \/ (rng g))) +* (x,y)) * f) +* (((id ((rng f) \/ (rng g))) +* (x,y)) * g) by A2, Th9, XBOOLE_1:7
.= (f +~ (x,y)) +* (g +~ (x,y)) by A1, Th116, XBOOLE_1:7 ;
::_thesis: verum
end;

definition
let a, b be set ;
funca followed_by b ->  set  equals :: FUNCT_7:def 13
(NAT --> b) +* (0,a);
coherence
 (NAT --> b) +* (0,a) is set ;
end;

:: deftheorem  defines followed_by FUNCT_7:def_13_:_
 for a, b being set holds a followed_by b = (NAT --> b) +* (0,a);

registration
let a, b be set ;
clustera followed_by b ->  Relation-like Function-like ;
coherence
 ( a followed_by b is Function-like & a followed_by b is Relation-like ) ;
end;


theorem Th118: :: FUNCT_7:118
for a, b being set holds dom (a followed_by b) = NAT
proof
let a, b be set ; ::_thesis: dom (a followed_by b) = NAT
thus  dom (a followed_by b) = dom ((NAT --> b) +* (0,a))
.= dom (NAT --> b) by Th30
.= NAT by FUNCOP_1:13 ; ::_thesis: verum
end;

definition
let X be non empty set ;
let a, b be Element of X;
:: original: followed_by
redefine funca followed_by b ->  sequence of X;
coherence
 a followed_by b is sequence of X
proof
 a followed_by b = (NAT --> b) +* (0,a) ;
hence  a followed_by b is sequence of X by Th115; ::_thesis: verum
end;
end;

theorem Th119: :: FUNCT_7:119
for a, b being set holds (a followed_by b) . 0 = a
proof
let a, b be set ; ::_thesis: (a followed_by b) . 0 = a
 dom (NAT --> b) = NAT by FUNCOP_1:13;
hence  (a followed_by b) . 0 = a by Th31; ::_thesis: verum
end;

theorem Th120: :: FUNCT_7:120
for a, b being set
for n being Nat st n > 0 holds
(a followed_by b) . n = b
proof
let a, b be set ; ::_thesis: for n being Nat st n > 0 holds
(a followed_by b) . n = b
let n be Nat; ::_thesis: ( n > 0 implies (a followed_by b) . n = b )
A1: n in NAT by ORDINAL1:def_12;
assume  n > 0 ; ::_thesis: (a followed_by b) . n = b
hence (a followed_by b) . n = (NAT --> b) . n by Th32
.= b by A1, FUNCOP_1:7 ;
::_thesis: verum
end;

definition
let a, b, c be set ;
func(a,b) followed_by c ->  set  equals :: FUNCT_7:def 14
(NAT --> c) +* ((0,1) --> (a,b));
coherence
 (NAT --> c) +* ((0,1) --> (a,b)) is set ;
end;

:: deftheorem  defines followed_by FUNCT_7:def_14_:_
 for a, b, c being set holds (a,b) followed_by c = (NAT --> c) +* ((0,1) --> (a,b));

registration
let a, b, c be set ;
cluster(a,b) followed_by c ->  Relation-like Function-like ;
coherence
 ( (a,b) followed_by c is Function-like & (a,b) followed_by c is Relation-like ) ;
end;

theorem Th121: :: FUNCT_7:121
for a, b, c being set holds dom ((a,b) followed_by c) = NAT
proof
let a, b, c be set ; ::_thesis: dom ((a,b) followed_by c) = NAT
thus  dom ((a,b) followed_by c) = (dom (NAT --> c)) \/ (dom ((0,1) --> (a,b))) by FUNCT_4:def_1
.= NAT \/ (dom ((0,1) --> (a,b))) by FUNCOP_1:13
.= NAT \/ {0,1} by FUNCT_4:62
.= NAT by XBOOLE_1:12 ; ::_thesis: verum
end;

theorem Th122: :: FUNCT_7:122
for a, b, c being set holds ((a,b) followed_by c) . 0 = a
proof
let a, b, c be set ; ::_thesis: ((a,b) followed_by c) . 0 = a
 dom ((0,1) --> (a,b)) = {0,1} by FUNCT_4:62;
then A1: 0 in dom ((0,1) --> (a,b)) by TARSKI:def_2;
thus ((a,b) followed_by c) . 0 = ((NAT --> c) +* ((0,1) --> (a,b))) . 0
.= ((0,1) --> (a,b)) . 0 by A1, FUNCT_4:13
.= a by FUNCT_4:63 ; ::_thesis: verum
end;

theorem Th123: :: FUNCT_7:123
for a, b, c being set holds ((a,b) followed_by c) . 1 = b
proof
let a, b, c be set ; ::_thesis: ((a,b) followed_by c) . 1 = b
 dom ((0,1) --> (a,b)) = {0,1} by FUNCT_4:62;
then A1: 1 in dom ((0,1) --> (a,b)) by TARSKI:def_2;
thus ((a,b) followed_by c) . 1 = ((NAT --> c) +* ((0,1) --> (a,b))) . 1
.= ((0,1) --> (a,b)) . 1 by A1, FUNCT_4:13
.= b by FUNCT_4:63 ; ::_thesis: verum
end;

theorem Th124: :: FUNCT_7:124
for a, b, c being set
for n being Nat st n > 1 holds
((a,b) followed_by c) . n = c
proof
let a, b, c be set ; ::_thesis: for n being Nat st n > 1 holds
((a,b) followed_by c) . n = c
let n be Nat; ::_thesis: ( n > 1 implies ((a,b) followed_by c) . n = c )
assume A1: n > 1 ; ::_thesis: ((a,b) followed_by c) . n = c
 dom ((0,1) --> (a,b)) = {0,1} by FUNCT_4:62;
then A2: not n in dom ((0,1) --> (a,b)) by A1, TARSKI:def_2;
A3: n in NAT by ORDINAL1:def_12;
thus ((a,b) followed_by c) . n = ((NAT --> c) +* ((0,1) --> (a,b))) . n
.= (NAT --> c) . n by A2, FUNCT_4:11
.= c by A3, FUNCOP_1:7 ; ::_thesis: verum
end;

theorem Th125: :: FUNCT_7:125
for a, b, c being set holds (a,b) followed_by c = (a followed_by c) +* (1,b)
proof
let a, b, c be set ; ::_thesis: (a,b) followed_by c = (a followed_by c) +* (1,b)
set f = (a,b) followed_by c;
set g = (a followed_by c) +* (1,b);
A1: dom (a followed_by c) = NAT by Th118;
A2: dom ((a,b) followed_by c) = NAT by Th121;
hence  dom ((a,b) followed_by c) = dom ((a followed_by c) +* (1,b)) by A1, Th30; :: according to FUNCT_1:def_11 ::_thesis: for b1 being set holds
( not b1 in proj1 ((a,b) followed_by c) or ((a,b) followed_by c) . b1 = ((a followed_by c) +* (1,b)) . b1 )
let x be set ; ::_thesis: ( not x in proj1 ((a,b) followed_by c) or ((a,b) followed_by c) . x = ((a followed_by c) +* (1,b)) . x )
assume  x in dom ((a,b) followed_by c) ; ::_thesis: ((a,b) followed_by c) . x = ((a followed_by c) +* (1,b)) . x
then reconsider n = x as Nat by A2;
percases ( n = 0 or n = 1 or n > 1 ) by NAT_1:25;
supposeA3: n = 0 ; ::_thesis: ((a,b) followed_by c) . x = ((a followed_by c) +* (1,b)) . x
hence ((a,b) followed_by c) . x = a by Th122
.= (a followed_by c) . x by A3, Th119
.= ((a followed_by c) +* (1,b)) . x by A3, Th32 ;
::_thesis: verum
end;
supposeA4: n = 1 ; ::_thesis: ((a,b) followed_by c) . x = ((a followed_by c) +* (1,b)) . x
hence ((a,b) followed_by c) . x = b by Th123
.= ((a followed_by c) +* (1,b)) . x by A1, A4, Th31 ;
::_thesis: verum
end;
supposeA5: n > 1 ; ::_thesis: ((a,b) followed_by c) . x = ((a followed_by c) +* (1,b)) . x
hence ((a,b) followed_by c) . x = c by Th124
.= (a followed_by c) . x by A5, Th120
.= ((a followed_by c) +* (1,b)) . x by A5, Th32 ;
::_thesis: verum
end;
end;
end;

definition
let X be non empty set ;
let a, b, c be Element of X;
:: original: followed_by
redefine func(a,b) followed_by c ->  sequence of X;
coherence
 (a,b) followed_by c is sequence of X
proof
 (a,b) followed_by c = (a followed_by c) +* (1,b) by Th125;
hence  (a,b) followed_by c is sequence of X by Th115; ::_thesis: verum
end;
end;

theorem Th126: :: FUNCT_7:126
for a, b being set holds rng (a followed_by b) = {a,b}
proof
let a, b be set ; ::_thesis: rng (a followed_by b) = {a,b}
 rng (NAT --> b) = {b} by FUNCOP_1:8;
then  rng ((NAT --> b) +* (0,a)) c= {b} \/ {a} by Th100;
hence  rng (a followed_by b) c= {a,b} by ENUMSET1:1; :: according to XBOOLE_0:def_10 ::_thesis: {a,b} c= rng (a followed_by b)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {a,b} or x in rng (a followed_by b) )
assume A1: x in {a,b} ; ::_thesis: x in rng (a followed_by b)
A2: dom (NAT --> b) = NAT by FUNCOP_1:13;
then  1 in dom (NAT --> b) ;
then A3: 1 in dom (a followed_by b) by Th30;
 (a followed_by b) . 1 = b by Th120;
then A4: b in rng (a followed_by b) by A3, FUNCT_1:3;
 a in rng (a followed_by b) by A2, Th102;
hence  x in rng (a followed_by b) by A1, A4, TARSKI:def_2; ::_thesis: verum
end;

theorem :: FUNCT_7:127
for a, b, c being set holds rng ((a,b) followed_by c) = {a,b,c}
proof
let a, b, c be set ; ::_thesis: rng ((a,b) followed_by c) = {a,b,c}
A1: (a,b) followed_by c = (a followed_by c) +* (1,b) by Th125;
then A2: dom ((a,b) followed_by c) = dom (a followed_by c) by Th30
.= dom (NAT --> c) by Th30
.= NAT by FUNCOP_1:13 ;
 ((a,b) followed_by c) . 2 = c by Th124;
then A3: c in rng ((a,b) followed_by c) by A2, FUNCT_1:3;
 ((a,b) followed_by c) . 1 = b by Th123;
then A4: b in rng ((a,b) followed_by c) by A2, FUNCT_1:3;
 rng (a followed_by c) = {a,c} by Th126;
then  rng ((a followed_by c) +* (1,b)) c= {a,c} \/ {b} by Th100;
then  rng ((a followed_by c) +* (1,b)) c= {a,c,b} by ENUMSET1:3;
hence  rng ((a,b) followed_by c) c= {a,b,c} by A1, ENUMSET1:57; :: according to XBOOLE_0:def_10 ::_thesis: {a,b,c} c= rng ((a,b) followed_by c)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {a,b,c} or x in rng ((a,b) followed_by c) )
 ((a,b) followed_by c) . 0 = a by Th122;
then A5: a in rng ((a,b) followed_by c) by A2, FUNCT_1:3;
assume  x in {a,b,c} ; ::_thesis: x in rng ((a,b) followed_by c)
hence  x in rng ((a,b) followed_by c) by A5, A4, A3, ENUMSET1:def_1; ::_thesis: verum
end;

definition
let A, B be set ;
let f be Function of A,B;
let x be set ;
let y be Element of B;
:: original: +*
redefine funcf +* (x,y) ->  Function of A,B;
coherence
 f +* (x,y) is Function of A,B
proof
set F = f +* (x,y);
A1: rng (f +* (x,y)) c= (rng f) \/ {y} by Th100;
percases ( B is empty or not B is empty ) ;
suppose B is empty ; ::_thesis: f +* (x,y) is Function of A,B
then  dom f is empty ;
hence  f +* (x,y) is Function of A,B by Def3; ::_thesis: verum
end;
supposeA2: not B is empty ; ::_thesis: f +* (x,y) is Function of A,B
then  {y} c= B by ZFMISC_1:31;
then  (rng f) \/ {y} c= B by XBOOLE_1:8;
then A3: rng (f +* (x,y)) c= B by A1, XBOOLE_1:1;
 dom f = A by A2, FUNCT_2:def_1;
then  dom (f +* (x,y)) = A by Th30;
hence  f +* (x,y) is Function of A,B by A2, A3, FUNCT_2:def_1, RELSET_1:4; ::_thesis: verum
end;
end;
end;
end;

theorem :: FUNCT_7:128
for A, B being non empty set
for f being Function of A,B
for x being Element of A
for y being set holds (f +* (x,y)) . x = y
proof
let A, B be non empty set ; ::_thesis: for f being Function of A,B
for x being Element of A
for y being set holds (f +* (x,y)) . x = y
let f be Function of A,B; ::_thesis: for x being Element of A
for y being set holds (f +* (x,y)) . x = y
let x be Element of A; ::_thesis: for y being set holds (f +* (x,y)) . x = y
let y be set ; ::_thesis: (f +* (x,y)) . x = y
 x in A ;
then  x in dom f by FUNCT_2:def_1;
hence  (f +* (x,y)) . x = y by Th31; ::_thesis: verum
end;

theorem :: FUNCT_7:129
for A, B being non empty set
for f, g being Function of A,B
for x being Element of A st ( for y being Element of A st f . y <> g . y holds
y = x ) holds
f = g +* (x,(f . x))
proof
let A, B be non empty set ; ::_thesis: for f, g being Function of A,B
for x being Element of A st ( for y being Element of A st f . y <> g . y holds
y = x ) holds
f = g +* (x,(f . x))
let f, g be Function of A,B; ::_thesis: for x being Element of A st ( for y being Element of A st f . y <> g . y holds
y = x ) holds
f = g +* (x,(f . x))
let x be Element of A; ::_thesis: ( ( for y being Element of A st f . y <> g . y holds
y = x ) implies f = g +* (x,(f . x)) )
assume A1: for y being Element of A st f . y <> g . y holds
y = x ; ::_thesis: f = g +* (x,(f . x))
 x in A ;
then A2: x in dom g by FUNCT_2:def_1;
A3: dom f = A by FUNCT_2:def_1
.= A \/ {x} by XBOOLE_1:12
.= (dom g) \/ {x} by FUNCT_2:def_1
.= (dom g) \/ (dom (x .--> (f . x))) by FUNCOP_1:13 ;
 for e being set st e in (dom g) \/ (dom (x .--> (f . x))) holds
( ( e in dom (x .--> (f . x)) implies f . e = (x .--> (f . x)) . e ) & ( not e in dom (x .--> (f . x)) implies f . e = g . e ) )
proof
let e be set ; ::_thesis: ( e in (dom g) \/ (dom (x .--> (f . x))) implies ( ( e in dom (x .--> (f . x)) implies f . e = (x .--> (f . x)) . e ) & ( not e in dom (x .--> (f . x)) implies f . e = g . e ) ) )
assume A4: e in (dom g) \/ (dom (x .--> (f . x))) ; ::_thesis: ( ( e in dom (x .--> (f . x)) implies f . e = (x .--> (f . x)) . e ) & ( not e in dom (x .--> (f . x)) implies f . e = g . e ) )
thus  ( e in dom (x .--> (f . x)) implies f . e = (x .--> (f . x)) . e ) ::_thesis: ( not e in dom (x .--> (f . x)) implies f . e = g . e )
proof
assume  e in dom (x .--> (f . x)) ; ::_thesis: f . e = (x .--> (f . x)) . e
then  e in {x} by FUNCOP_1:13;
then  e = x by TARSKI:def_1;
hence  f . e = (x .--> (f . x)) . e by FUNCOP_1:72; ::_thesis: verum
end;
assume  not e in dom (x .--> (f . x)) ; ::_thesis: f . e = g . e
then  not e in {x} by FUNCOP_1:13;
then  e <> x by TARSKI:def_1;
hence  f . e = g . e by A1, A4; ::_thesis: verum
end;
hence f = g +* (x .--> (f . x)) by A3, FUNCT_4:def_1
.= g +* (x,(f . x)) by A2, Def3 ;
::_thesis: verum
end;

theorem Th130: :: FUNCT_7:130
for A being set
for f being Function
for g being b1 -defined Function holds f,f +* g equal_outside A
proof
let A be set ; ::_thesis: for f being Function
for g being A -defined Function holds f,f +* g equal_outside A
let f be Function; ::_thesis: for g being A -defined Function holds f,f +* g equal_outside A
let g be A -defined Function; ::_thesis: f,f +* g equal_outside A
 dom g c= A ;
hence  f,f +* g equal_outside A by Th29; ::_thesis: verum
end;

theorem :: FUNCT_7:131
for A being set
for f, g being b1 -defined Function holds f,g equal_outside A
proof
let A be set ; ::_thesis: for f, g being A -defined Function holds f,g equal_outside A
let f, g be A -defined Function; ::_thesis: f,g equal_outside A
A1: (dom g) \ A = {} by XBOOLE_1:37;
 (dom f) \ A = {} by XBOOLE_1:37;
hence f | ((dom f) \ A) = {}
.= g | ((dom g) \ A) by A1 ;
:: according to FUNCT_7:def_2 ::_thesis: verum
end;

theorem :: FUNCT_7:132
for A being set
for f, g being b1 -defined Function
for h being Function holds h +* f,h +* g equal_outside A
proof
let A be set ; ::_thesis: for f, g being A -defined Function
for h being Function holds h +* f,h +* g equal_outside A
let f, g be A -defined Function; ::_thesis: for h being Function holds h +* f,h +* g equal_outside A
let h be Function; ::_thesis: h +* f,h +* g equal_outside A
 h,h +* f equal_outside A by Th130;
then A1: h +* f,h equal_outside A by Th26;
 h,h +* g equal_outside A by Th130;
hence  h +* f,h +* g equal_outside A by A1, Th27; ::_thesis: verum
end;

theorem :: FUNCT_7:133
for m being Element of NAT
for I being NAT -defined Function holds card (Shift (I,m)) = card I
proof
let m be Element of NAT ; ::_thesis: for I being NAT -defined Function holds card (Shift (I,m)) = card I
defpred S1[ set ] means verum;
deffunc H1( Element of NAT ) ->  Element of NAT = $1;
let I be NAT -defined Function; ::_thesis: card (Shift (I,m)) = card I
A1: for x being set st x in dom I holds
ex d being Element of NAT st x = H1(d) ;
defpred S2[ Element of NAT ] means $1 in dom I;
deffunc H2( Element of NAT ) ->  Element of NAT = $1 + m;
set B =  {  l where l is Element of NAT : H1(l) in dom I  }  ;
set C =  {  H2(l) where l is Element of NAT : ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] )  }  ;
set D =  {  H2(l) where l is Element of NAT : l in  {  l where l is Element of NAT : H1(l) in dom I  }   }  ;
set E =  {  (l + m) where l is Element of NAT : l in dom I  }  ;
A2: for d1, d2 being Element of NAT st H1(d1) = H1(d2) holds
d1 = d2 ;
A3: dom I, {  l where l is Element of NAT : H1(l) in dom I  }  are_equipotent from FUNCT_7:sch_3(A1, A2);
A4:  {  H2(l) where l is Element of NAT : ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] )  }  c=  {  (l + m) where l is Element of NAT : l in dom I  }
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in  {  H2(l) where l is Element of NAT : ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] )  }  or e in  {  (l + m) where l is Element of NAT : l in dom I  }  )
assume  e in  {  H2(l) where l is Element of NAT : ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] )  }  ; ::_thesis: e in  {  (l + m) where l is Element of NAT : l in dom I  }
then consider l being Element of NAT  such that
A5: e = H2(l) and
A6: ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] ) ;
 ex n being Element of NAT st
( n = l & S2[n] ) by A6;
hence  e in  {  (l + m) where l is Element of NAT : l in dom I  }  by A5; ::_thesis: verum
end;
set B =  {  l where l is Element of NAT : S2[l]  }  ;
  {  l where l is Element of NAT : S2[l]  }  is Subset of NAT from DOMAIN_1:sch_7();
then A7:  {  l where l is Element of NAT : S2[l]  }  c= NAT ;
set B =  {  l where l is Element of NAT : l in dom I  }  ;
A8: for d1, d2 being Element of NAT st H2(d1) = H2(d2) holds
d1 = d2 ;
A9:  {  l where l is Element of NAT : l in dom I  }  , {  H2(l) where l is Element of NAT : l in  {  l where l is Element of NAT : H1(l) in dom I  }   }  are_equipotent from FUNCT_7:sch_4(A7, A8);
A10:  {  (l + m) where l is Element of NAT : l in dom I  }  c=  {  H2(l) where l is Element of NAT : l in  {  l where l is Element of NAT : H1(l) in dom I  }   }
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in  {  (l + m) where l is Element of NAT : l in dom I  }  or e in  {  H2(l) where l is Element of NAT : l in  {  l where l is Element of NAT : H1(l) in dom I  }   }  )
assume  e in  {  (l + m) where l is Element of NAT : l in dom I  }  ; ::_thesis: e in  {  H2(l) where l is Element of NAT : l in  {  l where l is Element of NAT : H1(l) in dom I  }   }
then consider l being Element of NAT  such that
A11: e = l + m and
A12: l in dom I ;
 l in  {  l where l is Element of NAT : l in dom I  }  by A12;
hence  e in  {  H2(l) where l is Element of NAT : l in  {  l where l is Element of NAT : H1(l) in dom I  }   }  by A11; ::_thesis: verum
end;
A13: dom (Shift (I,m)) =  {  (l + m) where l is Element of NAT : l in dom I  }  by VALUED_1:def_12;
A14:  {  H2(l) where l is Element of NAT : l in  {  l where l is Element of NAT : H1(l) in dom I  }   }  c=  {  H2(l) where l is Element of NAT : ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] )  }
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in  {  H2(l) where l is Element of NAT : l in  {  l where l is Element of NAT : H1(l) in dom I  }   }  or e in  {  H2(l) where l is Element of NAT : ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] )  }  )
assume  e in  {  H2(l) where l is Element of NAT : l in  {  l where l is Element of NAT : H1(l) in dom I  }   }  ; ::_thesis: e in  {  H2(l) where l is Element of NAT : ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] )  }
then  ex l being Element of NAT st
( e = H2(l) & l in  {  l where l is Element of NAT : l in dom I  }  ) ;
hence  e in  {  H2(l) where l is Element of NAT : ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] )  }  ; ::_thesis: verum
end;
then   {  (l + m) where l is Element of NAT : l in dom I  }  c=  {  H2(l) where l is Element of NAT : ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] )  }  by A10, XBOOLE_1:1;
then A15:  {  H2(l) where l is Element of NAT : ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] )  }  =  {  (l + m) where l is Element of NAT : l in dom I  }  by A4, XBOOLE_0:def_10;
then   {  H2(l) where l is Element of NAT : ( l in  {  n where n is Element of NAT : S2[n]  }  & S1[l] )  }  =  {  H2(l) where l is Element of NAT : l in  {  l where l is Element of NAT : H1(l) in dom I  }   }  by A10, A14, XBOOLE_0:def_10;
then A16: dom (Shift (I,m)), dom I are_equipotent by A3, A15, A9, A13, WELLORD2:15;
thus  card (Shift (I,m)) = card (dom (Shift (I,m))) by CARD_1:62
.= card (dom I) by A16, CARD_1:5
.= card I by CARD_1:62 ; ::_thesis: verum
end;

theorem Th134: :: FUNCT_7:134
for A, B being set
for f, g being Function st dom f = dom g & dom f c= A \/ B & f | B = g | B holds
f,g equal_outside A
proof
let A, B be set ; ::_thesis: for f, g being Function st dom f = dom g & dom f c= A \/ B & f | B = g | B holds
f,g equal_outside A
let f, g be Function; ::_thesis: ( dom f = dom g & dom f c= A \/ B & f | B = g | B implies f,g equal_outside A )
assume that
A1: dom f = dom g and
A2: dom f c= A \/ B ; ::_thesis: ( not f | B = g | B or f,g equal_outside A )
assume  f | B = g | B ; ::_thesis: f,g equal_outside A
hence  f | ((dom f) \ A) = g | ((dom g) \ A) by A1, A2, RELAT_1:153, XBOOLE_1:43; :: according to FUNCT_7:def_2 ::_thesis: verum
end;

theorem Th135: :: FUNCT_7:135
for B, A being set
for f, g being Function st dom f = dom g & B c= dom f & A misses B & f,g equal_outside A holds
f | B = g | B
proof
let B, A be set ; ::_thesis: for f, g being Function st dom f = dom g & B c= dom f & A misses B & f,g equal_outside A holds
f | B = g | B
let f, g be Function; ::_thesis: ( dom f = dom g & B c= dom f & A misses B & f,g equal_outside A implies f | B = g | B )
assume that
A1: dom f = dom g and
A2: ( B c= dom f & A misses B ) ; ::_thesis: ( not f,g equal_outside A or f | B = g | B )
assume  f | ((dom f) \ A) = g | ((dom g) \ A) ; :: according to FUNCT_7:def_2 ::_thesis: f | B = g | B
hence  f | B = g | B by A1, A2, RELAT_1:153, XBOOLE_1:86; ::_thesis: verum
end;


theorem :: FUNCT_7:136
for I, A, B being set
for X, Y being ManySortedSet of I st I c= A \/ B & X | B = Y | B holds
X,Y equal_outside A
proof
let I, A, B be set ; ::_thesis: for X, Y being ManySortedSet of I st I c= A \/ B & X | B = Y | B holds
X,Y equal_outside A
let X, Y be ManySortedSet of I; ::_thesis: ( I c= A \/ B & X | B = Y | B implies X,Y equal_outside A )
 ( dom X = I & dom Y = I ) by PARTFUN1:def_2;
hence  ( I c= A \/ B & X | B = Y | B implies X,Y equal_outside A ) by Th134; ::_thesis: verum
end;

theorem :: FUNCT_7:137
for B, I, A being set
for X, Y being ManySortedSet of I st B c= I & A misses B & X,Y equal_outside A holds
X | B = Y | B
proof
let B, I, A be set ; ::_thesis: for X, Y being ManySortedSet of I st B c= I & A misses B & X,Y equal_outside A holds
X | B = Y | B
let X, Y be ManySortedSet of I; ::_thesis: ( B c= I & A misses B & X,Y equal_outside A implies X | B = Y | B )
 ( dom X = I & dom Y = I ) by PARTFUN1:def_2;
hence  ( B c= I & A misses B & X,Y equal_outside A implies X | B = Y | B ) by Th135; ::_thesis: verum
end;

registration
let V be non empty set ;
let f be V -valued Function;
let x be set ;
let y be Element of V;
clusterf +* (x,y) -> V -valued ;
coherence
 f +* (x,y) is V -valued
proof
A1: rng (f +* (x,y)) c= (rng f) \/ {y} by Th100;
thus  rng (f +* (x,y)) c= V by A1, XBOOLE_1:1; :: according to RELAT_1:def_19 ::_thesis: verum
end;
end;

theorem :: FUNCT_7:138
for f, g being Function
for x, y being set st f c= g & not x in dom f holds
f c= g +* (x,y)
proof
let f, g be Function; ::_thesis: for x, y being set st f c= g & not x in dom f holds
f c= g +* (x,y)
let x, y be set ; ::_thesis: ( f c= g & not x in dom f implies f c= g +* (x,y) )
assume A1: f c= g ; ::_thesis: ( x in dom f or f c= g +* (x,y) )
assume  not x in dom f ; ::_thesis: f c= g +* (x,y)
then A2: f c= f +* (x .--> y) by FUNCT_4:107;
percases ( x in dom g or not x in dom g ) ;
suppose x in dom g ; ::_thesis: f c= g +* (x,y)
then  g +* (x,y) = g +* (x .--> y) by Def3;
then  f +* (x .--> y) c= g +* (x,y) by A1, FUNCT_4:123;
hence  f c= g +* (x,y) by A2, XBOOLE_1:1; ::_thesis: verum
end;
suppose not x in dom g ; ::_thesis: f c= g +* (x,y)
hence  f c= g +* (x,y) by A1, Def3; ::_thesis: verum
end;
end;
end;

theorem :: FUNCT_7:139
for I being non empty set
for X being ManySortedSet of I
for l1, l2 being Element of I
for i1, i2 being set holds X +* ((l1,l2) --> (i1,i2)) = (X +* (l1,i1)) +* (l2,i2)
proof
let I be non empty set ; ::_thesis: for X being ManySortedSet of I
for l1, l2 being Element of I
for i1, i2 being set holds X +* ((l1,l2) --> (i1,i2)) = (X +* (l1,i1)) +* (l2,i2)
let X be ManySortedSet of I; ::_thesis: for l1, l2 being Element of I
for i1, i2 being set holds X +* ((l1,l2) --> (i1,i2)) = (X +* (l1,i1)) +* (l2,i2)
let l1, l2 be Element of I; ::_thesis: for i1, i2 being set holds X +* ((l1,l2) --> (i1,i2)) = (X +* (l1,i1)) +* (l2,i2)
let i1, i2 be set ; ::_thesis: X +* ((l1,l2) --> (i1,i2)) = (X +* (l1,i1)) +* (l2,i2)
A1: dom X = I by PARTFUN1:def_2;
then A2: l1 in dom X ;
 dom (X +* (l1,i1)) = I by A1, Th30;
then A3: l2 in dom (X +* (l1,i1)) ;
thus X +* ((l1,l2) --> (i1,i2)) = X +* ((l1 .--> i1) +* (l2 .--> i2))
.= (X +* (l1 .--> i1)) +* (l2 .--> i2) by FUNCT_4:14
.= (X +* (l1,i1)) +* (l2 .--> i2) by A2, Def3
.= (X +* (l1,i1)) +* (l2,i2) by A3, Def3 ; ::_thesis: verum
end;