let S be non empty non void ManySortedSign ; for U1, U2 being non-empty MSAlgebra of S
for F being ManySortedFunction of U1,U2 st F is_homomorphism U1,U2 holds
MSHomQuot F is_monomorphism QuotMSAlg U1,(MSCng F),U2
let U1, U2 be non-empty MSAlgebra of S; for F being ManySortedFunction of U1,U2 st F is_homomorphism U1,U2 holds
MSHomQuot F is_monomorphism QuotMSAlg U1,(MSCng F),U2
let F be ManySortedFunction of U1,U2; ( F is_homomorphism U1,U2 implies MSHomQuot F is_monomorphism QuotMSAlg U1,(MSCng F),U2 )
set mc = MSCng F;
set qa = QuotMSAlg U1,(MSCng F);
set qh = MSHomQuot F;
set S1 = the Sorts of U1;
assume A1:
F is_homomorphism U1,U2
; MSHomQuot F is_monomorphism QuotMSAlg U1,(MSCng F),U2
for o being OperSymbol of S st Args o,(QuotMSAlg U1,(MSCng F)) <> {} holds
for x being Element of Args o,(QuotMSAlg U1,(MSCng F)) holds ((MSHomQuot F) . (the_result_sort_of o)) . ((Den o,(QuotMSAlg U1,(MSCng F))) . x) = (Den o,U2) . ((MSHomQuot F) # x)
proof
let o be
OperSymbol of
S;
( Args o,(QuotMSAlg U1,(MSCng F)) <> {} implies for x being Element of Args o,(QuotMSAlg U1,(MSCng F)) holds ((MSHomQuot F) . (the_result_sort_of o)) . ((Den o,(QuotMSAlg U1,(MSCng F))) . x) = (Den o,U2) . ((MSHomQuot F) # x) )
assume
Args o,
(QuotMSAlg U1,(MSCng F)) <> {}
;
for x being Element of Args o,(QuotMSAlg U1,(MSCng F)) holds ((MSHomQuot F) . (the_result_sort_of o)) . ((Den o,(QuotMSAlg U1,(MSCng F))) . x) = (Den o,U2) . ((MSHomQuot F) # x)
let x be
Element of
Args o,
(QuotMSAlg U1,(MSCng F));
((MSHomQuot F) . (the_result_sort_of o)) . ((Den o,(QuotMSAlg U1,(MSCng F))) . x) = (Den o,U2) . ((MSHomQuot F) # x)
set ro =
the_result_sort_of o;
set ar =
the_arity_of o;
A2:
dom x = dom (the_arity_of o)
by MSUALG_3:6;
Args o,
(QuotMSAlg U1,(MSCng F)) = (((Class (MSCng F)) # ) * the Arity of S) . o
by MSUALG_1:def 9;
then consider a being
Element of
Args o,
U1 such that A3:
x = (MSCng F) # a
by Th2;
A4:
dom a = dom (the_arity_of o)
by MSUALG_3:6;
A5:
now let y be
set ;
( y in dom (the_arity_of o) implies ((MSHomQuot F) # x) . y = (F # a) . y )assume A6:
y in dom (the_arity_of o)
;
((MSHomQuot F) # x) . y = (F # a) . ythen reconsider n =
y as
Nat by ORDINAL1:def 13;
(the_arity_of o) . n in rng (the_arity_of o)
by A6, FUNCT_1:def 5;
then reconsider s =
(the_arity_of o) . n as
SortSymbol of
S ;
A7:
(the_arity_of o) /. n = (the_arity_of o) . n
by A6, PARTFUN1:def 8;
then
x . n = Class ((MSCng F) . s),
(a . n)
by A3, A6, Def9;
then A8:
x . n = Class (MSCng F,s),
(a . n)
by A1, Def20;
A9:
n in dom (the Sorts of U1 * (the_arity_of o))
by A6, PARTFUN1:def 4;
then
a . n in (the Sorts of U1 * (the_arity_of o)) . n
by MSUALG_3:6;
then reconsider an =
a . n as
Element of the
Sorts of
U1 . s by A9, FUNCT_1:22;
((MSHomQuot F) # x) . n =
((MSHomQuot F) . s) . (x . n)
by A2, A6, A7, MSUALG_3:def 8
.=
(MSHomQuot F,s) . (x . n)
by Def22
.=
(F . s) . an
by A1, A8, Def21
.=
(F # a) . n
by A4, A6, A7, MSUALG_3:def 8
;
hence
((MSHomQuot F) # x) . y = (F # a) . y
;
verum end;
o in the
carrier' of
S
;
then
o in dom (the Sorts of U1 * the ResultSort of S)
by PARTFUN1:def 4;
then A10:
(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
(
rng (Den o,U1) c= Result o,
U1 &
Result o,
U1 = the
Sorts of
U1 . (the_result_sort_of o) )
by MSUALG_1:def 10;
then
rng (Den o,U1) c= dom (QuotRes (MSCng F),o)
by A10, FUNCT_2:def 1;
then A11:
(
dom (Den o,U1) = Args o,
U1 &
dom ((QuotRes (MSCng F),o) * (Den o,U1)) = dom (Den o,U1) )
by FUNCT_2:def 1, RELAT_1:46;
the_arity_of o = the
Arity of
S . o
by MSUALG_1:def 6;
then A12:
product ((Class (MSCng F)) * (the_arity_of o)) = (((Class (MSCng F)) # ) * the Arity of S) . o
by MSAFREE:1;
reconsider da =
(Den o,U1) . a as
Element of the
Sorts of
U1 . (the_result_sort_of o) by A10, MSUALG_1:def 10;
A13:
(MSHomQuot F) . (the_result_sort_of o) = MSHomQuot F,
(the_result_sort_of o)
by Def22;
Den o,
(QuotMSAlg U1,(MSCng F)) =
(QuotCharact (MSCng F)) . o
by MSUALG_1:def 11
.=
QuotCharact (MSCng F),
o
by Def15
;
then (Den o,(QuotMSAlg U1,(MSCng F))) . x =
((QuotRes (MSCng F),o) * (Den o,U1)) . a
by A3, A12, Def14
.=
(QuotRes (MSCng F),o) . da
by A11, FUNCT_1:22
.=
Class (MSCng F),
da
by Def10
.=
Class (MSCng F,(the_result_sort_of o)),
da
by A1, Def20
;
then A14:
((MSHomQuot F) . (the_result_sort_of o)) . ((Den o,(QuotMSAlg U1,(MSCng F))) . x) =
(F . (the_result_sort_of o)) . ((Den o,U1) . a)
by A1, A13, Def21
.=
(Den o,U2) . (F # a)
by A1, MSUALG_3:def 9
;
(
dom ((MSHomQuot F) # x) = dom (the_arity_of o) &
dom (F # a) = dom (the_arity_of o) )
by MSUALG_3:6;
hence
((MSHomQuot F) . (the_result_sort_of o)) . ((Den o,(QuotMSAlg U1,(MSCng F))) . x) = (Den o,U2) . ((MSHomQuot F) # x)
by A5, A14, FUNCT_1:9;
verum
end;
hence
MSHomQuot F is_homomorphism QuotMSAlg U1,(MSCng F),U2
by MSUALG_3:def 9; MSUALG_3:def 11 MSHomQuot F is "1-1"
for i being set st i in the carrier of S holds
(MSHomQuot F) . i is one-to-one
proof
let i be
set ;
( i in the carrier of S implies (MSHomQuot F) . i is one-to-one )
set f =
(MSHomQuot F) . i;
assume
i in the
carrier of
S
;
(MSHomQuot F) . i is one-to-one
then reconsider s =
i as
SortSymbol of
S ;
A15:
(MSHomQuot F) . i = MSHomQuot F,
s
by Def22;
for
x1,
x2 being
set st
x1 in dom ((MSHomQuot F) . i) &
x2 in dom ((MSHomQuot F) . i) &
((MSHomQuot F) . i) . x1 = ((MSHomQuot F) . i) . x2 holds
x1 = x2
proof
let x1,
x2 be
set ;
( x1 in dom ((MSHomQuot F) . i) & x2 in dom ((MSHomQuot F) . i) & ((MSHomQuot F) . i) . x1 = ((MSHomQuot F) . i) . x2 implies x1 = x2 )
assume that A16:
x1 in dom ((MSHomQuot F) . i)
and A17:
x2 in dom ((MSHomQuot F) . i)
and A18:
((MSHomQuot F) . i) . x1 = ((MSHomQuot F) . i) . x2
;
x1 = x2
x1 in (Class (MSCng F)) . s
by A15, A16, FUNCT_2:def 1;
then
x1 in Class ((MSCng F) . s)
by Def8;
then consider a1 being
set such that A19:
a1 in the
Sorts of
U1 . s
and A20:
x1 = Class ((MSCng F) . s),
a1
by EQREL_1:def 5;
x2 in (Class (MSCng F)) . s
by A15, A17, FUNCT_2:def 1;
then
x2 in Class ((MSCng F) . s)
by Def8;
then consider a2 being
set such that A21:
a2 in the
Sorts of
U1 . s
and A22:
x2 = Class ((MSCng F) . s),
a2
by EQREL_1:def 5;
reconsider a2 =
a2 as
Element of the
Sorts of
U1 . s by A21;
A23:
(MSCng F) . s = MSCng F,
s
by A1, Def20;
then A24:
((MSHomQuot F) . i) . x2 = (F . s) . a2
by A1, A15, A22, Def21;
reconsider a1 =
a1 as
Element of the
Sorts of
U1 . s by A19;
((MSHomQuot F) . i) . x1 = (F . s) . a1
by A1, A15, A23, A20, Def21;
then
[a1,a2] in MSCng F,
s
by A18, A24, Def19;
hence
x1 = x2
by A23, A20, A22, EQREL_1:44;
verum
end;
hence
(MSHomQuot F) . i is
one-to-one
by FUNCT_1:def 8;
verum
end;
hence
MSHomQuot F is "1-1"
by MSUALG_3:1; verum