let S be locally_directed OrderSortedSign; :: thesis: for U1, U2 being non-empty OSAlgebra of S
for F being ManySortedFunction of U1,U2
for R being OSCongruence of U1 st F is_homomorphism U1,U2 & F is order-sorted & R c= OSCng F holds
( OSHomQuot F,R is_homomorphism QuotOSAlg U1,R,U2 & OSHomQuot F,R is order-sorted )
let U1, U2 be non-empty OSAlgebra of S; :: thesis: for F being ManySortedFunction of U1,U2
for R being OSCongruence of U1 st F is_homomorphism U1,U2 & F is order-sorted & R c= OSCng F holds
( OSHomQuot F,R is_homomorphism QuotOSAlg U1,R,U2 & OSHomQuot F,R is order-sorted )
let F be ManySortedFunction of U1,U2; :: thesis: for R being OSCongruence of U1 st F is_homomorphism U1,U2 & F is order-sorted & R c= OSCng F holds
( OSHomQuot F,R is_homomorphism QuotOSAlg U1,R,U2 & OSHomQuot F,R is order-sorted )
let R be OSCongruence of U1; :: thesis: ( F is_homomorphism U1,U2 & F is order-sorted & R c= OSCng F implies ( OSHomQuot F,R is_homomorphism QuotOSAlg U1,R,U2 & OSHomQuot F,R is order-sorted ) )
set mc = R;
set qa = QuotOSAlg U1,R;
set qh = OSHomQuot F,R;
set S1 = the Sorts of U1;
assume A1:
( F is_homomorphism U1,U2 & F is order-sorted & R c= OSCng F )
; :: thesis: ( OSHomQuot F,R is_homomorphism QuotOSAlg U1,R,U2 & OSHomQuot F,R is order-sorted )
for o being Element of the carrier' of S st Args o,(QuotOSAlg U1,R) <> {} holds
for x being Element of Args o,(QuotOSAlg U1,R) holds ((OSHomQuot F,R) . (the_result_sort_of o)) . ((Den o,(QuotOSAlg U1,R)) . x) = (Den o,U2) . ((OSHomQuot F,R) # x)
proof
let o be
Element of the
carrier' of
S;
:: thesis: ( Args o,(QuotOSAlg U1,R) <> {} implies for x being Element of Args o,(QuotOSAlg U1,R) holds ((OSHomQuot F,R) . (the_result_sort_of o)) . ((Den o,(QuotOSAlg U1,R)) . x) = (Den o,U2) . ((OSHomQuot F,R) # x) )
assume
Args o,
(QuotOSAlg U1,R) <> {}
;
:: thesis: for x being Element of Args o,(QuotOSAlg U1,R) holds ((OSHomQuot F,R) . (the_result_sort_of o)) . ((Den o,(QuotOSAlg U1,R)) . x) = (Den o,U2) . ((OSHomQuot F,R) # x)
let x be
Element of
Args o,
(QuotOSAlg U1,R);
:: thesis: ((OSHomQuot F,R) . (the_result_sort_of o)) . ((Den o,(QuotOSAlg U1,R)) . x) = (Den o,U2) . ((OSHomQuot F,R) # x)
reconsider o1 =
o as
OperSymbol of
S ;
set ro =
the_result_sort_of o;
set ar =
the_arity_of o;
A2:
Den o,
(QuotOSAlg U1,R) =
(OSQuotCharact R) . o
by MSUALG_1:def 11
.=
OSQuotCharact R,
o1
by Def21
;
Args o,
(QuotOSAlg U1,R) = (((OSClass R) # ) * the Arity of S) . o
by MSUALG_1:def 9;
then consider a being
Element of
Args o,
U1 such that A3:
x = R #_os a
by Th15;
A4:
(
dom (Den o,U1) = Args o,
U1 &
rng (Den o,U1) c= Result o,
U1 )
by FUNCT_2:def 1;
o in the
carrier' of
S
;
then
o in dom (the Sorts of U1 * the ResultSort of S)
by PARTFUN1:def 4;
then A5:
(the Sorts of U1 * the ResultSort of S) . o =
the
Sorts of
U1 . (the ResultSort of S . o)
by FUNCT_1:22
.=
the
Sorts of
U1 . (the_result_sort_of o)
by MSUALG_1:def 7
;
then A6:
Result o,
U1 = the
Sorts of
U1 . (the_result_sort_of o)
by MSUALG_1:def 10;
reconsider da =
(Den o,U1) . a as
Element of the
Sorts of
U1 . (the_result_sort_of o) by A5, MSUALG_1:def 10;
A7:
(OSHomQuot F,R) . (the_result_sort_of o) = OSHomQuot F,
R,
(the_result_sort_of o)
by Def30;
rng (Den o,U1) c= dom (OSQuotRes R,o)
by A4, A5, A6, FUNCT_2:def 1;
then A8:
dom ((OSQuotRes R,o) * (Den o,U1)) = dom (Den o,U1)
by RELAT_1:46;
A9:
(
dom ((OSHomQuot F,R) # x) = dom (the_arity_of o) &
dom (F # a) = dom (the_arity_of o) &
dom x = dom (the_arity_of o) &
dom a = dom (the_arity_of o) )
by MSUALG_3:6;
A10:
now let y be
set ;
:: thesis: ( y in dom (the_arity_of o) implies ((OSHomQuot F,R) # x) . y = (F # a) . y )assume A11:
y in dom (the_arity_of o)
;
:: thesis: ((OSHomQuot F,R) # x) . y = (F # a) . ythen reconsider n =
y as
Nat ;
A12:
(the_arity_of o) /. n = (the_arity_of o) . n
by A11, PARTFUN1:def 8;
(the_arity_of o) . n in rng (the_arity_of o)
by A11, FUNCT_1:def 5;
then reconsider s =
(the_arity_of o) . n as
SortSymbol of
S ;
consider an being
Element of the
Sorts of
U1 . s such that A13:
(
an = a . n &
x . n = OSClass R,
an )
by A3, A11, A12, Def15;
((OSHomQuot F,R) # x) . n =
((OSHomQuot F,R) . s) . (x . n)
by A9, A11, A12, MSUALG_3:def 8
.=
(OSHomQuot F,R,s) . (OSClass R,an)
by A13, Def30
.=
(F . s) . an
by A1, Def29
.=
(F # a) . n
by A9, A11, A12, A13, MSUALG_3:def 8
;
hence
((OSHomQuot F,R) # x) . y = (F # a) . y
;
:: thesis: verum end;
the_arity_of o = the
Arity of
S . o
by MSUALG_1:def 6;
then
product ((OSClass R) * (the_arity_of o)) = (((OSClass R) # ) * the Arity of S) . o
by MSAFREE:1;
then (Den o,(QuotOSAlg U1,R)) . x =
((OSQuotRes R,o) * (Den o,U1)) . a
by A2, A3, Def20
.=
(OSQuotRes R,o) . da
by A4, A8, FUNCT_1:22
.=
OSClass R,
da
by Def16
;
then ((OSHomQuot F,R) . (the_result_sort_of o)) . ((Den o,(QuotOSAlg U1,R)) . x) =
(F . (the_result_sort_of o)) . ((Den o,U1) . a)
by A1, A7, Def29
.=
(Den o,U2) . (F # a)
by A1, MSUALG_3:def 9
;
hence
((OSHomQuot F,R) . (the_result_sort_of o)) . ((Den o,(QuotOSAlg U1,R)) . x) = (Den o,U2) . ((OSHomQuot F,R) # x)
by A9, A10, FUNCT_1:9;
:: thesis: verum
end;
hence
OSHomQuot F,R is_homomorphism QuotOSAlg U1,R,U2
by MSUALG_3:def 9; :: thesis: OSHomQuot F,R is order-sorted
thus
OSHomQuot F,R is order-sorted
:: thesis: verumproof
let s1,
s2 be
Element of
S;
:: according to OSALG_3:def 1 :: thesis: ( not s1 <= s2 or for b1 being set holds
( not b1 in dom ((OSHomQuot F,R) . s1) or ( b1 in dom ((OSHomQuot F,R) . s2) & ((OSHomQuot F,R) . s1) . b1 = ((OSHomQuot F,R) . s2) . b1 ) ) )
assume A14:
s1 <= s2
;
:: thesis: for b1 being set holds
( not b1 in dom ((OSHomQuot F,R) . s1) or ( b1 in dom ((OSHomQuot F,R) . s2) & ((OSHomQuot F,R) . s1) . b1 = ((OSHomQuot F,R) . s2) . b1 ) )
let a1 be
set ;
:: thesis: ( not a1 in dom ((OSHomQuot F,R) . s1) or ( a1 in dom ((OSHomQuot F,R) . s2) & ((OSHomQuot F,R) . s1) . a1 = ((OSHomQuot F,R) . s2) . a1 ) )
assume A15:
a1 in dom ((OSHomQuot F,R) . s1)
;
:: thesis: ( a1 in dom ((OSHomQuot F,R) . s2) & ((OSHomQuot F,R) . s1) . a1 = ((OSHomQuot F,R) . s2) . a1 )
reconsider sqa = the
Sorts of
(QuotOSAlg U1,R) as
OrderSortedSet of ;
reconsider S1O = the
Sorts of
U1 as
OrderSortedSet of
by OSALG_1:17;
A16:
S1O . s1 c= S1O . s2
by A14, OSALG_1:def 18;
A17:
(
dom ((OSHomQuot F,R) . s1) = the
Sorts of
(QuotOSAlg U1,R) . s1 &
dom ((OSHomQuot F,R) . s2) = the
Sorts of
(QuotOSAlg U1,R) . s2 )
by FUNCT_2:def 1;
sqa . s1 c= sqa . s2
by A14, OSALG_1:def 18;
hence
a1 in dom ((OSHomQuot F,R) . s2)
by A15, A17;
:: thesis: ((OSHomQuot F,R) . s1) . a1 = ((OSHomQuot F,R) . s2) . a1
a1 in (OSClass R) . s1
by A15;
then
a1 in OSClass R,
s1
by Def13;
then consider x being
set such that A18:
x in the
Sorts of
U1 . s1
and A19:
a1 = Class (CompClass R,(CComp s1)),
x
by Def12;
reconsider x1 =
x as
Element of the
Sorts of
U1 . s1 by A18;
reconsider x2 =
x as
Element of the
Sorts of
U1 . s2 by A16, A18;
reconsider s3 =
s1,
s4 =
s2 as
Element of
S ;
x1 in dom (F . s3)
by A18, FUNCT_2:def 1;
then A20:
(
x1 in dom (F . s4) &
(F . s3) . x1 = (F . s4) . x1 )
by A1, A14, OSALG_3:def 1;
A21:
a1 = OSClass R,
x2
by A14, A19, Th5;
thus ((OSHomQuot F,R) . s1) . a1 =
(OSHomQuot F,R,s1) . (OSClass R,x1)
by A19, Def30
.=
(F . s2) . x1
by A1, A20, Def29
.=
(OSHomQuot F,R,s2) . (OSClass R,x2)
by A1, Def29
.=
((OSHomQuot F,R) . s2) . a1
by A21, Def30
;
:: thesis: verum
end;