:: MSUALG_9 semantic presentation
theorem Th1: :: MSUALG_9:1
theorem Th2: :: MSUALG_9:2
theorem Th3: :: MSUALG_9:3
theorem Th4: :: MSUALG_9:4
theorem Th5: :: MSUALG_9:5
theorem Th6: :: MSUALG_9:6
theorem Th7: :: MSUALG_9:7
theorem Th8: :: MSUALG_9:8
theorem Th9: :: MSUALG_9:9
theorem Th10: :: MSUALG_9:10
theorem Th11: :: MSUALG_9:11
theorem Th12: :: MSUALG_9:12
theorem Th13: :: MSUALG_9:13
theorem Th14: :: MSUALG_9:14
theorem Th15: :: MSUALG_9:15
theorem Th16: :: MSUALG_9:16
theorem Th17: :: MSUALG_9:17
theorem Th18: :: MSUALG_9:18
theorem Th19: :: MSUALG_9:19
theorem Th20: :: MSUALG_9:20
definition
let I be
set ;
let A be
ManySortedSet of
I;
let B be
V5 ManySortedSet of
I,
C be
V5 ManySortedSet of
I;
let F be
ManySortedFunction of
A,
[|B,C|];
func Mpr1 c5 -> ManySortedFunction of
a2,
a3 means :
Def1:
:: MSUALG_9:def 1
for
i being
set st
i in I holds
it . i = pr1 (F . i);
existence
ex b1 being ManySortedFunction of A,B st
for i being set st i in I holds
b1 . i = pr1 (F . i)
uniqueness
for b1, b2 being ManySortedFunction of A,B st ( for i being set st i in I holds
b1 . i = pr1 (F . i) ) & ( for i being set st i in I holds
b2 . i = pr1 (F . i) ) holds
b1 = b2
func Mpr2 c5 -> ManySortedFunction of
a2,
a4 means :
Def2:
:: MSUALG_9:def 2
for
i being
set st
i in I holds
it . i = pr2 (F . i);
existence
ex b1 being ManySortedFunction of A,C st
for i being set st i in I holds
b1 . i = pr2 (F . i)
uniqueness
for b1, b2 being ManySortedFunction of A,C st ( for i being set st i in I holds
b1 . i = pr2 (F . i) ) & ( for i being set st i in I holds
b2 . i = pr2 (F . i) ) holds
b1 = b2
end;
:: deftheorem Def1 defines Mpr1 MSUALG_9:def 1 :
:: deftheorem Def2 defines Mpr2 MSUALG_9:def 2 :
theorem Th21: :: MSUALG_9:21
theorem Th22: :: MSUALG_9:22
theorem Th23: :: MSUALG_9:23
theorem Th24: :: MSUALG_9:24
theorem Th25: :: MSUALG_9:25
theorem Th26: :: MSUALG_9:26
theorem Th27: :: MSUALG_9:27
theorem Th28: :: MSUALG_9:28
theorem Th29: :: MSUALG_9:29
theorem Th30: :: MSUALG_9:30
theorem Th31: :: MSUALG_9:31
theorem Th32: :: MSUALG_9:32
theorem Th33: :: MSUALG_9:33
theorem Th34: :: MSUALG_9:34
E146:
now
let S be non
empty non
void ManySortedSign ;
let A be
non-empty MSAlgebra of
S;
let C1 be
MSCongruence of
A,
C2 be
MSCongruence of
A;
let G be
ManySortedFunction of
(QuotMSAlg A,C1),
(QuotMSAlg A,C2);
assume E39:
for
i being
Element of
S for
x being
Element of the
Sorts of
(QuotMSAlg A,C1) . i for
xx being
Element of the
Sorts of
A . i st
x = Class C1,
xx holds
(G . i) . x = Class C2,
xx
;
thus
G is
"onto"
proof
set sL = the
Sorts of
(QuotMSAlg A,C1);
set sP = the
Sorts of
(QuotMSAlg A,C2);
let i be
set ;
:: according to MSUALG_3:def 3
assume
i in the
carrier of
S
;
then reconsider s =
i as
SortSymbol of
S ;
rng (G . s) c= the
Sorts of
(QuotMSAlg A,C2) . s
;
hence
rng (G . i) c= the
Sorts of
(QuotMSAlg A,C2) . i
;
:: according to XBOOLE_0:def 10
let q be
set ;
:: according to TARSKI:def 3
assume E42:
q in the
Sorts of
(QuotMSAlg A,C2) . i
;
q in Class (C2 . s)
by , MSUALG_4:def 8;
then consider a being
set such that E43:
a in the
Sorts of
A . s
and E47:
q = Class (C2 . s),
a
by EQREL_1:def 5;
reconsider a =
a as
Element of the
Sorts of
A . s by ;
E48:
dom (G . s) = the
Sorts of
(QuotMSAlg A,C1) . s
by FUNCT_2:def 1;
Class (C1 . s),
a in Class (C1 . s)
by EQREL_1:def 5;
then
Class C1,
a in Class (C1 . s)
;
then reconsider x =
Class C1,
a as
Element of the
Sorts of
(QuotMSAlg A,C1) . s by MSUALG_4:def 8;
(G . s) . x =
Class C2,
a
by
.=
Class (C2 . s),
a
;
hence
q in rng (G . i)
by , , FUNCT_1:def 5;
end;
end;
theorem Th35: :: MSUALG_9:35
theorem Th36: :: MSUALG_9:36
for
S being non
empty non
void ManySortedSign for
A being
non-empty MSAlgebra of
S for
C1,
C2 being
MSCongruence of
A for
G being
ManySortedFunction of
(QuotMSAlg A,C1),
(QuotMSAlg A,C2) st ( for
i being
Element of
S for
x being
Element of the
Sorts of
(QuotMSAlg A,C1) . i for
xx being
Element of the
Sorts of
A . i st
x = Class C1,
xx holds
(G . i) . x = Class C2,
xx ) holds
G is_epimorphism QuotMSAlg A,
C1,
QuotMSAlg A,
C2
theorem Th37: :: MSUALG_9:37