set M = MSAlgCat S,A;
set G = MSAlgCat S,A;
set GM = the Arrows of (MSAlgCat S,A);
set C = the Comp of (MSAlgCat S,A);
thus
MSAlgCat S,A is transitive
( MSAlgCat S,A is associative & MSAlgCat S,A is with_units )proof
let o1,
o2,
o3 be
object of ;
ALTCAT_1:def 4 ( <^o1,o2^> = {} or <^o2,o3^> = {} or not <^o1,o3^> = {} )
reconsider o1' =
o1,
o2' =
o2,
o3' =
o3 as
Element of
MSAlg_set S,
A by Def4;
assume that A1:
<^o1,o2^> <> {}
and A2:
<^o2,o3^> <> {}
;
not <^o1,o3^> = {}
consider t being
Element of
MSAlg_morph S,
A,
o1',
o2';
MSAlg_morph S,
A,
o1',
o2' <> {}
by A1, Def4;
then consider M,
N being
strict feasible MSAlgebra of
S,
f being
ManySortedFunction of ,
M such that A3:
M = o1'
and A4:
N = o2'
and
f = t
and A5:
the
Sorts of
M is_transformable_to the
Sorts of
N
and A6:
f is_homomorphism M,
N
by Def3;
consider s being
Element of
MSAlg_morph S,
A,
o2',
o3';
MSAlg_morph S,
A,
o2',
o3' <> {}
by A2, Def4;
then consider M1,
N1 being
strict feasible MSAlgebra of
S,
g being
ManySortedFunction of ,
M1 such that A7:
M1 = o2'
and A8:
N1 = o3'
and
g = s
and A9:
the
Sorts of
M1 is_transformable_to the
Sorts of
N1
and A10:
g is_homomorphism M1,
N1
by Def3;
reconsider g' =
g as
ManySortedFunction of ,
N by A4, A7;
consider GF being
ManySortedFunction of ,
M such that
GF = g' ** f
and A11:
GF is_homomorphism M,
N1
by A4, A5, A6, A7, A9, A10, Th5;
the
Sorts of
M is_transformable_to the
Sorts of
N1
by A4, A5, A7, A9, AUTALG_1:11;
then
GF in MSAlg_morph S,
A,
o1',
o3'
by A3, A8, A11, Def3;
hence
not
<^o1,o3^> = {}
by Def4;
verum
end;
thus
the Comp of (MSAlgCat S,A) is associative
ALTCAT_1:def 17 MSAlgCat S,A is with_units proof
let i,
j,
k,
l be
Element of ;
ALTCAT_1:def 7 for b1, b2, b3 being set holds
( not b1 in the Arrows of (MSAlgCat S,A) . i,j or not b2 in the Arrows of (MSAlgCat S,A) . j,k or not b3 in the Arrows of (MSAlgCat S,A) . k,l or (the Comp of (MSAlgCat S,A) . i,k,l) . b3,((the Comp of (MSAlgCat S,A) . i,j,k) . b2,b1) = (the Comp of (MSAlgCat S,A) . i,j,l) . ((the Comp of (MSAlgCat S,A) . j,k,l) . b3,b2),b1 )let f,
g,
h be
set ;
( not f in the Arrows of (MSAlgCat S,A) . i,j or not g in the Arrows of (MSAlgCat S,A) . j,k or not h in the Arrows of (MSAlgCat S,A) . k,l or (the Comp of (MSAlgCat S,A) . i,k,l) . h,((the Comp of (MSAlgCat S,A) . i,j,k) . g,f) = (the Comp of (MSAlgCat S,A) . i,j,l) . ((the Comp of (MSAlgCat S,A) . j,k,l) . h,g),f )
reconsider i' =
i,
j' =
j,
k' =
k,
l' =
l as
Element of
MSAlg_set S,
A by Def4;
assume that A12:
f in the
Arrows of
(MSAlgCat S,A) . i,
j
and A13:
g in the
Arrows of
(MSAlgCat S,A) . j,
k
and A14:
h in the
Arrows of
(MSAlgCat S,A) . k,
l
;
(the Comp of (MSAlgCat S,A) . i,k,l) . h,((the Comp of (MSAlgCat S,A) . i,j,k) . g,f) = (the Comp of (MSAlgCat S,A) . i,j,l) . ((the Comp of (MSAlgCat S,A) . j,k,l) . h,g),f
g in MSAlg_morph S,
A,
j',
k'
by A13, Def4;
then consider M2,
N2 being
strict feasible MSAlgebra of
S,
G being
ManySortedFunction of ,
M2 such that A15:
M2 = j'
and A16:
N2 = k'
and A17:
g = G
and A18:
the
Sorts of
M2 is_transformable_to the
Sorts of
N2
and A19:
G is_homomorphism M2,
N2
by Def3;
h in MSAlg_morph S,
A,
k',
l'
by A14, Def4;
then consider M3,
N3 being
strict feasible MSAlgebra of
S,
H being
ManySortedFunction of ,
M3 such that A20:
M3 = k'
and A21:
N3 = l'
and A22:
h = H
and A23:
the
Sorts of
M3 is_transformable_to the
Sorts of
N3
and A24:
H is_homomorphism M3,
N3
by Def3;
reconsider G' =
G as
ManySortedFunction of ,
M2 by A16, A20;
consider HG being
ManySortedFunction of ,
M2 such that A25:
HG = H ** G'
and A26:
HG is_homomorphism M2,
N3
by A16, A18, A19, A20, A23, A24, Th5;
A27:
(the Comp of (MSAlgCat S,A) . j,k,l) . H,
G = H ** G'
by A13, A14, A17, A22, Def4;
f in MSAlg_morph S,
A,
i',
j'
by A12, Def4;
then consider M1,
N1 being
strict feasible MSAlgebra of
S,
F being
ManySortedFunction of ,
M1 such that A28:
M1 = i'
and A29:
N1 = j'
and A30:
f = F
and A31:
the
Sorts of
M1 is_transformable_to the
Sorts of
N1
and A32:
F is_homomorphism M1,
N1
by Def3;
A33:
(the Comp of (MSAlgCat S,A) . i,j,k) . g,
f = G ** F
by A12, A13, A30, A17, Def4;
consider GF being
ManySortedFunction of ,
M1 such that A34:
GF = G' ** F
and A35:
GF is_homomorphism M1,
M3
by A29, A31, A32, A15, A16, A18, A19, A20, Th5;
the
Sorts of
M1 is_transformable_to the
Sorts of
M3
by A29, A31, A15, A16, A18, A20, AUTALG_1:11;
then
G' ** F in MSAlg_morph S,
A,
i',
k'
by A28, A20, A34, A35, Def3;
then
GF in the
Arrows of
(MSAlgCat S,A) . i,
k
by A34, Def4;
then A36:
(the Comp of (MSAlgCat S,A) . i,k,l) . H,
GF = H ** GF
by A14, A22, Def4;
the
Sorts of
M2 is_transformable_to the
Sorts of
N3
by A16, A18, A20, A23, AUTALG_1:11;
then
HG in MSAlg_morph S,
A,
j',
l'
by A15, A21, A26, Def3;
then A37:
H ** G' in the
Arrows of
(MSAlgCat S,A) . j,
l
by A25, Def4;
(H ** G') ** F = H ** (G' ** F)
by PBOOLE:154;
hence
(the Comp of (MSAlgCat S,A) . i,k,l) . h,
((the Comp of (MSAlgCat S,A) . i,j,k) . g,f) = (the Comp of (MSAlgCat S,A) . i,j,l) . ((the Comp of (MSAlgCat S,A) . j,k,l) . h,g),
f
by A12, A30, A17, A22, A33, A34, A36, A27, A37, Def4;
verum
end;
thus
the Comp of (MSAlgCat S,A) is with_left_units
ALTCAT_1:def 18 the Comp of (MSAlgCat S,A) is with_right_units proof
let j be
Element of ;
ALTCAT_1:def 9 ex b1 being set st
( b1 in the Arrows of (MSAlgCat S,A) . j,j & ( for b2 being Element of the carrier of (MSAlgCat S,A)
for b3 being set holds
( not b3 in the Arrows of (MSAlgCat S,A) . b2,j or (the Comp of (MSAlgCat S,A) . b2,j,j) . b1,b3 = b3 ) ) )
reconsider j' =
j as
Element of
MSAlg_set S,
A by Def4;
consider MS being
strict feasible MSAlgebra of
S such that A38:
MS = j'
and
for
C being
Component of holds
C c= A
by Def2;
reconsider e =
id the
Sorts of
MS as
ManySortedFunction of ,
MS ;
take
e
;
( e in the Arrows of (MSAlgCat S,A) . j,j & ( for b1 being Element of the carrier of (MSAlgCat S,A)
for b2 being set holds
( not b2 in the Arrows of (MSAlgCat S,A) . b1,j or (the Comp of (MSAlgCat S,A) . b1,j,j) . e,b2 = b2 ) ) )
(
e is_homomorphism MS,
MS & the
Arrows of
(MSAlgCat S,A) . j,
j = MSAlg_morph S,
A,
j',
j' )
by Def4, MSUALG_3:9;
hence A39:
e in the
Arrows of
(MSAlgCat S,A) . j,
j
by A38, Def3;
for b1 being Element of the carrier of (MSAlgCat S,A)
for b2 being set holds
( not b2 in the Arrows of (MSAlgCat S,A) . b1,j or (the Comp of (MSAlgCat S,A) . b1,j,j) . e,b2 = b2 )
let i be
Element of ;
for b1 being set holds
( not b1 in the Arrows of (MSAlgCat S,A) . i,j or (the Comp of (MSAlgCat S,A) . i,j,j) . e,b1 = b1 )
reconsider i' =
i as
Element of
MSAlg_set S,
A by Def4;
let f be
set ;
( not f in the Arrows of (MSAlgCat S,A) . i,j or (the Comp of (MSAlgCat S,A) . i,j,j) . e,f = f )
reconsider I =
i,
J =
j as
object of ;
assume A40:
f in the
Arrows of
(MSAlgCat S,A) . i,
j
;
(the Comp of (MSAlgCat S,A) . i,j,j) . e,f = f
then
f in MSAlg_morph S,
A,
i',
j'
by Def4;
then consider M1,
N1 being
strict feasible MSAlgebra of
S,
F being
ManySortedFunction of ,
M1 such that
M1 = i'
and A41:
N1 = j'
and A42:
F = f
and
the
Sorts of
M1 is_transformable_to the
Sorts of
N1
and
F is_homomorphism M1,
N1
by Def3;
reconsider F =
F as
ManySortedFunction of ,
M1 by A38, A41;
(the Comp of (MSAlgCat S,A) . I,J,J) . e,
f = e ** F
by A39, A40, A42, Def4;
hence
(the Comp of (MSAlgCat S,A) . i,j,j) . e,
f = f
by A42, MSUALG_3:4;
verum
end;
thus
the Comp of (MSAlgCat S,A) is with_right_units
verumproof
let i be
Element of ;
ALTCAT_1:def 8 ex b1 being set st
( b1 in the Arrows of (MSAlgCat S,A) . i,i & ( for b2 being Element of the carrier of (MSAlgCat S,A)
for b3 being set holds
( not b3 in the Arrows of (MSAlgCat S,A) . i,b2 or (the Comp of (MSAlgCat S,A) . i,i,b2) . b3,b1 = b3 ) ) )
reconsider i' =
i as
Element of
MSAlg_set S,
A by Def4;
consider MS being
strict feasible MSAlgebra of
S such that A43:
MS = i'
and
for
C being
Component of holds
C c= A
by Def2;
reconsider e =
id the
Sorts of
MS as
ManySortedFunction of ,
MS ;
take
e
;
( e in the Arrows of (MSAlgCat S,A) . i,i & ( for b1 being Element of the carrier of (MSAlgCat S,A)
for b2 being set holds
( not b2 in the Arrows of (MSAlgCat S,A) . i,b1 or (the Comp of (MSAlgCat S,A) . i,i,b1) . b2,e = b2 ) ) )
(
e is_homomorphism MS,
MS & the
Arrows of
(MSAlgCat S,A) . i,
i = MSAlg_morph S,
A,
i',
i' )
by Def4, MSUALG_3:9;
hence A44:
e in the
Arrows of
(MSAlgCat S,A) . i,
i
by A43, Def3;
for b1 being Element of the carrier of (MSAlgCat S,A)
for b2 being set holds
( not b2 in the Arrows of (MSAlgCat S,A) . i,b1 or (the Comp of (MSAlgCat S,A) . i,i,b1) . b2,e = b2 )
let j be
Element of ;
for b1 being set holds
( not b1 in the Arrows of (MSAlgCat S,A) . i,j or (the Comp of (MSAlgCat S,A) . i,i,j) . b1,e = b1 )
reconsider j' =
j as
Element of
MSAlg_set S,
A by Def4;
let f be
set ;
( not f in the Arrows of (MSAlgCat S,A) . i,j or (the Comp of (MSAlgCat S,A) . i,i,j) . f,e = f )
reconsider I =
i,
J =
j as
object of ;
assume A45:
f in the
Arrows of
(MSAlgCat S,A) . i,
j
;
(the Comp of (MSAlgCat S,A) . i,i,j) . f,e = f
then
f in MSAlg_morph S,
A,
i',
j'
by Def4;
then consider M1,
N1 being
strict feasible MSAlgebra of
S,
F being
ManySortedFunction of ,
M1 such that A46:
M1 = i'
and
N1 = j'
and A47:
F = f
and
the
Sorts of
M1 is_transformable_to the
Sorts of
N1
and
F is_homomorphism M1,
N1
by Def3;
reconsider F =
F as
ManySortedFunction of ,
MS by A43, A46;
(the Comp of (MSAlgCat S,A) . I,I,J) . f,
e = F ** e
by A44, A45, A47, Def4;
hence
(the Comp of (MSAlgCat S,A) . i,i,j) . f,
e = f
by A47, MSUALG_3:3;
verum
end;