:: Retracts and Inheritance
:: by Grzegorz Bancerek
::
:: Received September 7, 1999
:: Copyright (c) 1999-2021 Association of Mizar Users

theorem :: YELLOW16:1

canceled;

::$CT

theorem :: YELLOW16:2

for X being set
for L being non empty RelStr
for S being non empty SubRelStr of L
for f, g being Function of X, the carrier of S
for f9, g9 being Function of X, the carrier of L st f9 = f & g9 = g & f <= g holds
f9 <= g9

proof end;

theorem :: YELLOW16:3

for X being set
for L being non empty RelStr
for S being non empty full SubRelStr of L
for f, g being Function of X, the carrier of S
for f9, g9 being Function of X, the carrier of L st f9 = f & g9 = g & f9 <= g9 holds
f <= g

proof end;

registration

let S be non empty RelStr ;
let T be non empty reflexive antisymmetric RelStr ;

cluster Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V24( the carrier of S) quasi_total monotone directed-sups-preserving for Element of bool [: the carrier of S, the carrier of T:];

existence
ex b₁ being Function of S,T st
( b₁ is directed-sups-preserving & b₁ is monotone )

proof end;

end;

theorem :: YELLOW16:4

for f, g being Function st f is idempotent & rng g c= rng f & rng g c= dom f holds
f * g = g

proof end;

registration

let S be 1-sorted ;

cluster Relation-like the carrier of S -defined the carrier of S -valued Function-like V24( the carrier of S) quasi_total idempotent for Element of bool [: the carrier of S, the carrier of S:];

existence
ex b₁ being Function of S,S st b₁ is idempotent

proof end;

end;

theorem Th4: :: YELLOW16:5

for L being non empty up-complete Poset
for S being non empty full directed-sups-inheriting SubRelStr of L holds S is up-complete

proof end;

theorem Th5: :: YELLOW16:6

for L being non empty up-complete Poset
for f being Function of L,L st f is idempotent & f is directed-sups-preserving holds
Image f is directed-sups-inheriting

proof end;

theorem Th6: :: YELLOW16:7

for T being non empty RelStr
for S being non empty SubRelStr of T
for f being Function of T,S st f * (incl (S,T)) = id S holds
f is idempotent Function of T,T

proof end;

definition

let S, T be non empty Poset;
let f be Function;

pred f is_a_retraction_of T,S means :: YELLOW16:def 1
( f is directed-sups-preserving Function of T,S & f | the carrier of S = id S & S is full directed-sups-inheriting SubRelStr of T );

pred f is_an_UPS_retraction_of T,S means :: YELLOW16:def 2
( f is directed-sups-preserving Function of T,S & ex g being directed-sups-preserving Function of S,T st f * g = id S );

end;

:: deftheorem defines is_a_retraction_of YELLOW16:def 1 :
for S, T being non empty Poset
for f being Function holds
( f is_a_retraction_of T,S iff ( f is directed-sups-preserving Function of T,S & f | the carrier of S = id S & S is full directed-sups-inheriting SubRelStr of T ) );

:: deftheorem defines is_an_UPS_retraction_of YELLOW16:def 2 :
for S, T being non empty Poset
for f being Function holds
( f is_an_UPS_retraction_of T,S iff ( f is directed-sups-preserving Function of T,S & ex g being directed-sups-preserving Function of S,T st f * g = id S ) );

definition

let S, T be non empty Poset;

pred S is_a_retract_of T means :: YELLOW16:def 3
ex f being Function of T,S st f is_a_retraction_of T,S;

pred S is_an_UPS_retract_of T means :: YELLOW16:def 4
ex f being Function of T,S st f is_an_UPS_retraction_of T,S;

end;

:: deftheorem defines is_a_retract_of YELLOW16:def 3 :
for S, T being non empty Poset holds
( S is_a_retract_of T iff ex f being Function of T,S st f is_a_retraction_of T,S );

:: deftheorem defines is_an_UPS_retract_of YELLOW16:def 4 :
for S, T being non empty Poset holds
( S is_an_UPS_retract_of T iff ex f being Function of T,S st f is_an_UPS_retraction_of T,S );

theorem Th7: :: YELLOW16:8

for S, T being non empty Poset
for f being Function st f is_a_retraction_of T,S holds
f * (incl (S,T)) = id S

proof end;

theorem Th8: :: YELLOW16:9

for S being non empty Poset
for T being non empty up-complete Poset
for f being Function st f is_a_retraction_of T,S holds
f is_an_UPS_retraction_of T,S

proof end;

theorem Th9: :: YELLOW16:10

for S, T being non empty Poset
for f being Function st f is_a_retraction_of T,S holds
rng f = the carrier of S

proof end;

theorem Th10: :: YELLOW16:11

for S, T being non empty Poset
for f being Function st f is_an_UPS_retraction_of T,S holds
rng f = the carrier of S

proof end;

theorem Th11: :: YELLOW16:12

for S, T being non empty Poset
for f being Function st f is_a_retraction_of T,S holds
f is idempotent Function of T,T

proof end;

theorem Th12: :: YELLOW16:13

for T, S being non empty Poset
for f being Function of T,T st f is_a_retraction_of T,S holds
Image f = RelStr(# the carrier of S, the InternalRel of S #)

proof end;

theorem Th13: :: YELLOW16:14

for T being non empty up-complete Poset
for S being non empty Poset
for f being Function of T,T st f is_a_retraction_of T,S holds
( f is directed-sups-preserving & f is projection )

proof end;

theorem Th14: :: YELLOW16:15

for S, T being non empty reflexive transitive RelStr
for f being Function of S,T holds
( f is isomorphic iff ( f is monotone & ex g being monotone Function of T,S st
( f * g = id T & g * f = id S ) ) )

proof end;

theorem Th15: :: YELLOW16:16

for S, T being non empty Poset holds
( S,T are_isomorphic iff ex f being monotone Function of S,T ex g being monotone Function of T,S st
( f * g = id T & g * f = id S ) )

proof end;

theorem :: YELLOW16:17

for S, T being non empty up-complete Poset st S,T are_isomorphic holds
( S is_an_UPS_retract_of T & T is_an_UPS_retract_of S )

proof end;

theorem Th17: :: YELLOW16:18

for T, S being non empty Poset
for f being monotone Function of T,S
for g being monotone Function of S,T st f * g = id S holds
ex h being projection Function of T,T st
( h = g * f & h | the carrier of (Image h) = id (Image h) & S, Image h are_isomorphic )

proof end;

theorem Th18: :: YELLOW16:19

for T being non empty up-complete Poset
for S being non empty Poset
for f being Function st f is_an_UPS_retraction_of T,S holds
ex h being directed-sups-preserving projection Function of T,T st
( h is_a_retraction_of T, Image h & S, Image h are_isomorphic )

proof end;

theorem Th19: :: YELLOW16:20

for L being non empty up-complete Poset
for S being non empty Poset st S is_a_retract_of L holds
S is up-complete

proof end;

theorem Th20: :: YELLOW16:21

for L being non empty complete Poset
for S being non empty Poset st S is_a_retract_of L holds
S is complete

proof end;

theorem Th21: :: YELLOW16:22

for L being complete continuous LATTICE
for S being non empty Poset st S is_a_retract_of L holds
S is continuous

proof end;

theorem :: YELLOW16:23

for L being non empty up-complete Poset
for S being non empty Poset st S is_an_UPS_retract_of L holds
S is up-complete

proof end;

theorem :: YELLOW16:24

for L being non empty complete Poset
for S being non empty Poset st S is_an_UPS_retract_of L holds
S is complete

proof end;

theorem :: YELLOW16:25

for L being complete continuous LATTICE
for S being non empty Poset st S is_an_UPS_retract_of L holds
S is continuous

proof end;

theorem Th25: :: YELLOW16:26

for L being RelStr
for S being full SubRelStr of L
for R being SubRelStr of S holds
( R is full iff R is full SubRelStr of L )

proof end;

theorem :: YELLOW16:27

for L being non empty transitive RelStr
for S being non empty full directed-sups-inheriting SubRelStr of L
for R being non empty directed-sups-inheriting SubRelStr of S holds R is directed-sups-inheriting SubRelStr of L

proof end;

theorem :: YELLOW16:28

for L being non empty up-complete Poset
for S1, S2 being non empty full directed-sups-inheriting SubRelStr of L st S1 is SubRelStr of S2 holds
S1 is full directed-sups-inheriting SubRelStr of S2

proof end;

definition

let R be Relation;

attr R is Poset-yielding means :Def5: :: YELLOW16:def 5
for x being set st x in rng R holds
x is Poset;

end;

:: deftheorem Def5 defines Poset-yielding YELLOW16:def 5 :
for R being Relation holds
( R is Poset-yielding iff for x being set st x in rng R holds
x is Poset );

registration

cluster Relation-like Poset-yielding -> RelStr-yielding reflexive-yielding for set ;

coherence
for b₁ being Relation st b₁ is Poset-yielding holds
( b₁ is RelStr-yielding & b₁ is reflexive-yielding )

proof end;

end;

definition

let X be non empty set ;
let L be non empty RelStr ;
let i be Element of X;
let Y be Subset of (L |^ X);
:: original: pi
redefine func pi (Y,i) -> Subset of L;
coherence
pi (Y,i) is Subset of L

proof end;

end;

registration

let X be set ;
let S be Poset;

cluster X --> S -> Poset-yielding ;

coherence
X --> S is Poset-yielding

proof end;

end;

registration

let I be set ;

cluster Relation-like I -defined Function-like V24(I) non-Empty Poset-yielding for set ;

existence
ex b₁ being ManySortedSet of I st
( b₁ is Poset-yielding & b₁ is non-Empty )

proof end;

end;

registration

let I be non empty set ;
let J be non-Empty Poset-yielding ManySortedSet of I;

cluster product J -> transitive antisymmetric ;

coherence
( product J is transitive & product J is antisymmetric )

proof end;

end;

definition

let I be non empty set ;
let J be non-Empty Poset-yielding ManySortedSet of I;
let i be Element of I;
:: original: .
redefine func J . i -> non empty Poset;
coherence
J . i is non empty Poset

proof end;

end;

Lm1: now :: thesis: for I being non empty set
for J being non-Empty Poset-yielding ManySortedSet of I
for X being Subset of (product J) st ( for i being Element of I holds ex_sup_of pi (X,i),J . i ) holds
ex f being Element of (product J) st
( ( for i being Element of I holds f . i = sup (pi (X,i)) ) & f is_>=_than X & ( for g being Element of (product J) st X is_<=_than g holds
f <= g ) )

let I be non empty set ; :: thesis: for J being non-Empty Poset-yielding ManySortedSet of I
for X being Subset of (product J) st ( for i being Element of I holds ex_sup_of pi (X,i),J . i ) holds
ex f being Element of (product J) st
( ( for i being Element of I holds f . i = sup (pi (X,i)) ) & f is_>=_than X & ( for g being Element of (product J) st X is_<=_than g holds
f <= g ) )
let J be non-Empty Poset-yielding ManySortedSet of I; :: thesis: for X being Subset of (product J) st ( for i being Element of I holds ex_sup_of pi (X,i),J . i ) holds
ex f being Element of (product J) st
( ( for i being Element of I holds f . i = sup (pi (X,i)) ) & f is_>=_than X & ( for g being Element of (product J) st X is_<=_than g holds
f <= g ) )
let X be Subset of (product J); :: thesis: ( ( for i being Element of I holds ex_sup_of pi (X,i),J . i ) implies ex f being Element of (product J) st
( ( for i being Element of I holds f . i = sup (pi (X,i)) ) & f is_>=_than X & ( for g being Element of (product J) st X is_<=_than g holds
f <= g ) ) )
deffunc H₁( Element of I) -> Element of the carrier of (J . $1) = sup (pi (X,$1));
consider f being ManySortedSet of I such that
A1: for i being Element of I holds f . i = H₁(i) from PBOOLE:sch 5();

A2: now :: thesis: for i being Element of I holds f . i is Element of (J . i)

let i be Element of I; :: thesis: f . i is Element of (J . i)
f . i = sup (pi (X,i)) by A1;
hence f . i is Element of (J . i) ; :: thesis: verum

end;

dom f = I by PARTFUN1:def 2;
then reconsider f = f as Element of (product J) by A2, WAYBEL_3:27;
assume A3: for i being Element of I holds ex_sup_of pi (X,i),J . i ; :: thesis: ex f being Element of (product J) st
( ( for i being Element of I holds f . i = sup (pi (X,i)) ) & f is_>=_than X & ( for g being Element of (product J) st X is_<=_than g holds
f <= g ) )
take f = f; :: thesis: ( ( for i being Element of I holds f . i = sup (pi (X,i)) ) & f is_>=_than X & ( for g being Element of (product J) st X is_<=_than g holds
f <= g ) )
thus for i being Element of I holds f . i = sup (pi (X,i)) by A1; :: thesis: ( f is_>=_than X & ( for g being Element of (product J) st X is_<=_than g holds
f <= g ) )
thus f is_>=_than X :: thesis: for g being Element of (product J) st X is_<=_than g holds
f <= g

proof

let x be Element of (product J); :: according to LATTICE3:def 9 :: thesis: ( not x in X or x <= f )
assume A4: x in X ; :: thesis: x <= f

now :: thesis: for i being Element of I holds x . i <= f . i

let i be Element of I; :: thesis: x . i <= f . i
A5: f . i = sup (pi (X,i)) by A1;
A6: x . i in pi (X,i) by A4, CARD_3:def 6;
ex_sup_of pi (X,i),J . i by A3;
then f . i is_>=_than pi (X,i) by A5, YELLOW_0:30;
hence x . i <= f . i by A6; :: thesis: verum

end;

hence x <= f by WAYBEL_3:28; :: thesis: verum

end;

let g be Element of (product J); :: thesis: ( X is_<=_than g implies f <= g )
assume A7: X is_<=_than g ; :: thesis: f <= g

now :: thesis: for i being Element of I holds f . i <= g . i

let i be Element of I; :: thesis: f . i <= g . i
A8: ex_sup_of pi (X,i),J . i by A3;
A9: g . i is_>=_than pi (X,i)

proof

let a be Element of (J . i); :: according to LATTICE3:def 9 :: thesis: ( not a in pi (X,i) or a <= g . i )
assume a in pi (X,i) ; :: thesis: a <= g . i
then consider h being Function such that
A10: h in X and
A11: a = h . i by CARD_3:def 6;
reconsider h = h as Element of (product J) by A10;
h <= g by A7, A10;
hence a <= g . i by A11, WAYBEL_3:28; :: thesis: verum

end;

f . i = sup (pi (X,i)) by A1;
hence f . i <= g . i by A8, A9, YELLOW_0:30; :: thesis: verum

end;

hence f <= g by WAYBEL_3:28; :: thesis: verum

end;

Lm2: now :: thesis: for I being non empty set
for J being non-Empty Poset-yielding ManySortedSet of I
for X being Subset of (product J) st ( for i being Element of I holds ex_inf_of pi (X,i),J . i ) holds
ex f being Element of (product J) st
( ( for i being Element of I holds f . i = inf (pi (X,i)) ) & f is_<=_than X & ( for g being Element of (product J) st X is_>=_than g holds
f >= g ) )

let I be non empty set ; :: thesis: for J being non-Empty Poset-yielding ManySortedSet of I
for X being Subset of (product J) st ( for i being Element of I holds ex_inf_of pi (X,i),J . i ) holds
ex f being Element of (product J) st
( ( for i being Element of I holds f . i = inf (pi (X,i)) ) & f is_<=_than X & ( for g being Element of (product J) st X is_>=_than g holds
f >= g ) )
let J be non-Empty Poset-yielding ManySortedSet of I; :: thesis: for X being Subset of (product J) st ( for i being Element of I holds ex_inf_of pi (X,i),J . i ) holds
ex f being Element of (product J) st
( ( for i being Element of I holds f . i = inf (pi (X,i)) ) & f is_<=_than X & ( for g being Element of (product J) st X is_>=_than g holds
f >= g ) )
let X be Subset of (product J); :: thesis: ( ( for i being Element of I holds ex_inf_of pi (X,i),J . i ) implies ex f being Element of (product J) st
( ( for i being Element of I holds f . i = inf (pi (X,i)) ) & f is_<=_than X & ( for g being Element of (product J) st X is_>=_than g holds
f >= g ) ) )
deffunc H₁( Element of I) -> Element of the carrier of (J . $1) = inf (pi (X,$1));
consider f being ManySortedSet of I such that
A1: for i being Element of I holds f . i = H₁(i) from PBOOLE:sch 5();

A2: now :: thesis: for i being Element of I holds f . i is Element of (J . i)

let i be Element of I; :: thesis: f . i is Element of (J . i)
f . i = inf (pi (X,i)) by A1;
hence f . i is Element of (J . i) ; :: thesis: verum

end;

dom f = I by PARTFUN1:def 2;
then reconsider f = f as Element of (product J) by A2, WAYBEL_3:27;
assume A3: for i being Element of I holds ex_inf_of pi (X,i),J . i ; :: thesis: ex f being Element of (product J) st
( ( for i being Element of I holds f . i = inf (pi (X,i)) ) & f is_<=_than X & ( for g being Element of (product J) st X is_>=_than g holds
f >= g ) )
take f = f; :: thesis: ( ( for i being Element of I holds f . i = inf (pi (X,i)) ) & f is_<=_than X & ( for g being Element of (product J) st X is_>=_than g holds
f >= g ) )
thus for i being Element of I holds f . i = inf (pi (X,i)) by A1; :: thesis: ( f is_<=_than X & ( for g being Element of (product J) st X is_>=_than g holds
f >= g ) )
thus f is_<=_than X :: thesis: for g being Element of (product J) st X is_>=_than g holds
f >= g

proof

let x be Element of (product J); :: according to LATTICE3:def 8 :: thesis: ( not x in X or f <= x )
assume A4: x in X ; :: thesis: f <= x

now :: thesis: for i being Element of I holds x . i >= f . i

let i be Element of I; :: thesis: x . i >= f . i
A5: f . i = inf (pi (X,i)) by A1;
A6: x . i in pi (X,i) by A4, CARD_3:def 6;
ex_inf_of pi (X,i),J . i by A3;
then f . i is_<=_than pi (X,i) by A5, YELLOW_0:31;
hence x . i >= f . i by A6; :: thesis: verum

end;

hence f <= x by WAYBEL_3:28; :: thesis: verum

end;

let g be Element of (product J); :: thesis: ( X is_>=_than g implies f >= g )
assume A7: X is_>=_than g ; :: thesis: f >= g

now :: thesis: for i being Element of I holds f . i >= g . i

let i be Element of I; :: thesis: f . i >= g . i
A8: ex_inf_of pi (X,i),J . i by A3;
A9: g . i is_<=_than pi (X,i)

proof

let a be Element of (J . i); :: according to LATTICE3:def 8 :: thesis: ( not a in pi (X,i) or g . i <= a )
assume a in pi (X,i) ; :: thesis: g . i <= a
then consider h being Function such that
A10: h in X and
A11: a = h . i by CARD_3:def 6;
reconsider h = h as Element of (product J) by A10;
h >= g by A7, A10;
hence a >= g . i by A11, WAYBEL_3:28; :: thesis: verum

end;

f . i = inf (pi (X,i)) by A1;
hence f . i >= g . i by A8, A9, YELLOW_0:31; :: thesis: verum

end;

hence f >= g by WAYBEL_3:28; :: thesis: verum

end;

theorem Th28: :: YELLOW16:29

for I being non empty set
for J being non-Empty Poset-yielding ManySortedSet of I
for f being Element of (product J)
for X being Subset of (product J) holds
( f is_>=_than X iff for i being Element of I holds f . i is_>=_than pi (X,i) )

proof end;

theorem Th29: :: YELLOW16:30

for I being non empty set
for J being non-Empty Poset-yielding ManySortedSet of I
for f being Element of (product J)
for X being Subset of (product J) holds
( f is_<=_than X iff for i being Element of I holds f . i is_<=_than pi (X,i) )

proof end;

theorem Th30: :: YELLOW16:31

for I being non empty set
for J being non-Empty Poset-yielding ManySortedSet of I
for X being Subset of (product J) holds
( ex_sup_of X, product J iff for i being Element of I holds ex_sup_of pi (X,i),J . i )

proof end;

theorem Th31: :: YELLOW16:32

for I being non empty set
for J being non-Empty Poset-yielding ManySortedSet of I
for X being Subset of (product J) holds
( ex_inf_of X, product J iff for i being Element of I holds ex_inf_of pi (X,i),J . i )

proof end;

theorem Th32: :: YELLOW16:33

for I being non empty set
for J being non-Empty Poset-yielding ManySortedSet of I
for X being Subset of (product J) st ex_sup_of X, product J holds
for i being Element of I holds (sup X) . i = sup (pi (X,i))

proof end;

theorem Th33: :: YELLOW16:34

for I being non empty set
for J being non-Empty Poset-yielding ManySortedSet of I
for X being Subset of (product J) st ex_inf_of X, product J holds
for i being Element of I holds (inf X) . i = inf (pi (X,i))

proof end;

theorem Th34: :: YELLOW16:35

for I being non empty set
for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I
for X being directed Subset of (product J)
for i being Element of I holds pi (X,i) is directed

proof end;

theorem Th35: :: YELLOW16:36

for I being non empty set
for J, K being RelStr-yielding non-Empty ManySortedSet of I st ( for i being Element of I holds K . i is SubRelStr of J . i ) holds
product K is SubRelStr of product J

proof end;

theorem Th36: :: YELLOW16:37

for I being non empty set
for J, K being RelStr-yielding non-Empty ManySortedSet of I st ( for i being Element of I holds K . i is full SubRelStr of J . i ) holds
product K is full SubRelStr of product J

proof end;

theorem :: YELLOW16:38

for L being non empty RelStr
for S being non empty SubRelStr of L
for X being set holds S |^ X is SubRelStr of L |^ X

proof end;

theorem Th38: :: YELLOW16:39

for L being non empty RelStr
for S being non empty full SubRelStr of L
for X being set holds S |^ X is full SubRelStr of L |^ X

proof end;

definition

let S, T be non empty RelStr ;
let X be set ;

pred S inherits_sup_of X,T means :: YELLOW16:def 6
( ex_sup_of X,T implies "\/" (X,T) in the carrier of S );

pred S inherits_inf_of X,T means :: YELLOW16:def 7
( ex_inf_of X,T implies "/\" (X,T) in the carrier of S );

end;

:: deftheorem defines inherits_sup_of YELLOW16:def 6 :
for S, T being non empty RelStr
for X being set holds
( S inherits_sup_of X,T iff ( ex_sup_of X,T implies "\/" (X,T) in the carrier of S ) );

:: deftheorem defines inherits_inf_of YELLOW16:def 7 :
for S, T being non empty RelStr
for X being set holds
( S inherits_inf_of X,T iff ( ex_inf_of X,T implies "/\" (X,T) in the carrier of S ) );

theorem :: YELLOW16:40

for T being non empty transitive RelStr
for S being non empty full SubRelStr of T
for X being Subset of S holds
( S inherits_sup_of X,T iff ( ex_sup_of X,T implies ( ex_sup_of X,S & sup X = "\/" (X,T) ) ) ) by YELLOW_0:64;

theorem :: YELLOW16:41

for T being non empty transitive RelStr
for S being non empty full SubRelStr of T
for X being Subset of S holds
( S inherits_inf_of X,T iff ( ex_inf_of X,T implies ( ex_inf_of X,S & inf X = "/\" (X,T) ) ) ) by YELLOW_0:63;

scheme :: YELLOW16:sch 1
ProductSupsInheriting{ F₁() -> non empty set , F₂() -> non-Empty Poset-yielding ManySortedSet of F₁(), F₃() -> non-Empty Poset-yielding ManySortedSet of F₁(), P₁[ set , set ] } :

for X being Subset of (product F₃()) st P₁[X, product F₃()] holds
product F₃() inherits_sup_of X, product F₂()

provided

A1: for X being Subset of (product F₃()) st P₁[X, product F₃()] holds
for i being Element of F₁() holds P₁[ pi (X,i),F₃() . i] and
A2: for i being Element of F₁() holds F₃() . i is full SubRelStr of F₂() . i and
A3: for i being Element of F₁()
for X being Subset of (F₃() . i) st P₁[X,F₃() . i] holds
F₃() . i inherits_sup_of X,F₂() . i

proof end;

scheme :: YELLOW16:sch 2
PowerSupsInheriting{ F₁() -> non empty set , F₂() -> non empty Poset, F₃() -> non empty full SubRelStr of F₂(), P₁[ set , set ] } :

for X being Subset of (F₃() |^ F₁()) st P₁[X,F₃() |^ F₁()] holds
F₃() |^ F₁() inherits_sup_of X,F₂() |^ F₁()

provided

A1: for X being Subset of (F₃() |^ F₁()) st P₁[X,F₃() |^ F₁()] holds
for i being Element of F₁() holds P₁[ pi (X,i),F₃()] and
A2: for X being Subset of F₃() st P₁[X,F₃()] holds
F₃() inherits_sup_of X,F₂()

proof end;

scheme :: YELLOW16:sch 3
ProductInfsInheriting{ F₁() -> non empty set , F₂() -> non-Empty Poset-yielding ManySortedSet of F₁(), F₃() -> non-Empty Poset-yielding ManySortedSet of F₁(), P₁[ set , set ] } :

for X being Subset of (product F₃()) st P₁[X, product F₃()] holds
product F₃() inherits_inf_of X, product F₂()

provided

A1: for X being Subset of (product F₃()) st P₁[X, product F₃()] holds
for i being Element of F₁() holds P₁[ pi (X,i),F₃() . i] and
A2: for i being Element of F₁() holds F₃() . i is full SubRelStr of F₂() . i and
A3: for i being Element of F₁()
for X being Subset of (F₃() . i) st P₁[X,F₃() . i] holds
F₃() . i inherits_inf_of X,F₂() . i

proof end;

scheme :: YELLOW16:sch 4
PowerInfsInheriting{ F₁() -> non empty set , F₂() -> non empty Poset, F₃() -> non empty full SubRelStr of F₂(), P₁[ set , set ] } :

for X being Subset of (F₃() |^ F₁()) st P₁[X,F₃() |^ F₁()] holds
F₃() |^ F₁() inherits_inf_of X,F₂() |^ F₁()

provided

A1: for X being Subset of (F₃() |^ F₁()) st P₁[X,F₃() |^ F₁()] holds
for i being Element of F₁() holds P₁[ pi (X,i),F₃()] and
A2: for X being Subset of F₃() st P₁[X,F₃()] holds
F₃() inherits_inf_of X,F₂()

proof end;

registration

let I be set ;
let L be non empty RelStr ;
let X be non empty Subset of (L |^ I);
let i be set ;

cluster pi (X,i) -> non empty ;

coherence
not pi (X,i) is empty

proof end;

end;

theorem :: YELLOW16:42

for L being non empty Poset
for S being non empty full directed-sups-inheriting SubRelStr of L
for X being non empty set holds S |^ X is directed-sups-inheriting SubRelStr of L |^ X

proof end;

registration

let I be non empty set ;
let J be RelStr-yielding non-Empty ManySortedSet of I;
let X be non empty Subset of (product J);
let i be set ;

cluster pi (X,i) -> non empty ;

coherence
not pi (X,i) is empty

proof end;

end;

theorem Th42: :: YELLOW16:43

for X being non empty set
for L being non empty up-complete Poset holds L |^ X is up-complete

proof end;

registration

let L be non empty up-complete Poset;
let X be non empty set ;

cluster L |^ X -> up-complete ;

coherence
L |^ X is up-complete by Th42;

end;

theorem Th43: :: YELLOW16:44

for T being non empty TopSpace
for S being non empty SubSpace of T
for f being Function of T,S st f is being_a_retraction holds
rng f = the carrier of S

proof end;

theorem :: YELLOW16:45

for T being non empty TopSpace
for S being non empty SubSpace of T
for f being continuous Function of T,S st f is being_a_retraction holds
f is idempotent

proof end;

theorem Th45: :: YELLOW16:46

for T being non empty TopSpace
for V being open Subset of T holds chi (V, the carrier of T) is continuous Function of T,Sierpinski_Space

proof end;

theorem :: YELLOW16:47

for T being non empty TopSpace
for x, y being Element of T st ( for W being open Subset of T st y in W holds
x in W ) holds
(0,1) --> (y,x) is continuous Function of Sierpinski_Space,T

proof end;

theorem :: YELLOW16:48

for T being non empty TopSpace
for x, y being Element of T
for V being open Subset of T st x in V & not y in V holds
(chi (V, the carrier of T)) * ((0,1) --> (y,x)) = id Sierpinski_Space

proof end;

theorem :: YELLOW16:49

for T being non empty 1-sorted
for V, W being Subset of T
for S being TopAugmentation of BoolePoset {0}
for f, g being Function of T,S st f = chi (V, the carrier of T) & g = chi (W, the carrier of T) holds
( V c= W iff f <= g )

proof end;

theorem :: YELLOW16:50

for L being non empty RelStr
for X being non empty set
for R being non empty full SubRelStr of L |^ X st ( for a being set holds
( a is Element of R iff ex x being Element of L st a = X --> x ) ) holds
L,R are_isomorphic

proof end;

theorem :: YELLOW16:51

for S, T being non empty TopSpace holds
( S,T are_homeomorphic iff ex f being continuous Function of S,T ex g being continuous Function of T,S st
( f * g = id T & g * f = id S ) )

proof end;

theorem Th51: :: YELLOW16:52

for T, S, R being non empty TopSpace
for f being Function of T,S
for g being Function of S,T
for h being Function of S,R st f * g = id S & h is being_homeomorphism holds
(h * f) * (g * (h ")) = id R

proof end;

theorem Th52: :: YELLOW16:53

for T, S, R being non empty TopSpace st S is_Retract_of T & S,R are_homeomorphic holds
R is_Retract_of T

proof end;

theorem Th53: :: YELLOW16:54

for T being non empty TopSpace
for S being non empty SubSpace of T holds incl (S,T) is continuous

proof end;

theorem Th54: :: YELLOW16:55

for T being non empty TopSpace
for S being non empty SubSpace of T
for f being continuous Function of T,S st f is being_a_retraction holds
f * (incl (S,T)) = id S

proof end;

theorem Th55: :: YELLOW16:56

for T being non empty TopSpace
for S being non empty SubSpace of T st S is_a_retract_of T holds
S is_Retract_of T

proof end;

theorem :: YELLOW16:57

for R, T being non empty TopSpace holds
( R is_Retract_of T iff ex S being non empty SubSpace of T st
( S is_a_retract_of T & S,R are_homeomorphic ) )

proof end;