let A, B be non empty transitive AltCatStr ; ( A,B are_opposite & A is associative implies B is associative )
assume that
A1:
A,B are_opposite
and
A2:
A is associative
; B is associative
deffunc H1( set , set , set , set , set ) -> set = (the Comp of A . $3,$2,$1) . $4,$5;
A3:
now let a,
b,
c be
object of ;
( <^a,b^> <> {} & <^b,c^> <> {} implies for f being Morphism of ,
for g being Morphism of , holds g * f = H1(a,b,c,f,g) )assume that A4:
<^a,b^> <> {}
and A5:
<^b,c^> <> {}
;
for f being Morphism of ,
for g being Morphism of , holds g * f = H1(a,b,c,f,g)let f be
Morphism of ,;
for g being Morphism of , holds g * f = H1(a,b,c,f,g)let g be
Morphism of ,;
g * f = H1(a,b,c,f,g)reconsider a' =
a,
b' =
b,
c' =
c as
object of
by A1, Th6;
A6:
<^a,b^> = <^b',a'^>
by A1, Th7;
A7:
<^b,c^> = <^c',b'^>
by A1, Th7;
reconsider f' =
f as
Morphism of ,
by A1, Th7;
reconsider g' =
g as
Morphism of ,
by A1, Th7;
thus g * f =
f' * g'
by A1, A4, A5, Th9
.=
H1(
a,
b,
c,
f,
g)
by A4, A5, A6, A7, ALTCAT_1:def 10
;
verum end;
A8:
now let a,
b,
c,
d be
object of ;
for f, g, h being set st f in <^a,b^> & g in <^b,c^> & h in <^c,d^> holds
H1(a,c,d,H1(a,b,c,f,g),h) = H1(a,b,d,f,H1(b,c,d,g,h))let f,
g,
h be
set ;
( f in <^a,b^> & g in <^b,c^> & h in <^c,d^> implies H1(a,c,d,H1(a,b,c,f,g),h) = H1(a,b,d,f,H1(b,c,d,g,h)) )reconsider a' =
a,
b' =
b,
c' =
c,
d' =
d as
object of
by A1, Def3;
assume A9:
f in <^a,b^>
;
( g in <^b,c^> & h in <^c,d^> implies H1(a,c,d,H1(a,b,c,f,g),h) = H1(a,b,d,f,H1(b,c,d,g,h)) )then A10:
f in <^b',a'^>
by A1, Th9;
reconsider f' =
f as
Morphism of ,
by A1, A9, Th9;
assume A11:
g in <^b,c^>
;
( h in <^c,d^> implies H1(a,c,d,H1(a,b,c,f,g),h) = H1(a,b,d,f,H1(b,c,d,g,h)) )then A12:
g in <^c',b'^>
by A1, Th9;
reconsider g' =
g as
Morphism of ,
by A1, A11, Th9;
assume A13:
h in <^c,d^>
;
H1(a,c,d,H1(a,b,c,f,g),h) = H1(a,b,d,f,H1(b,c,d,g,h))then A14:
h in <^d',c'^>
by A1, Th9;
reconsider h' =
h as
Morphism of ,
by A1, A13, Th9;
A15:
<^c',a'^> <> {}
by A10, A12, ALTCAT_1:def 4;
A16:
<^d',b'^> <> {}
by A12, A14, ALTCAT_1:def 4;
thus H1(
a,
c,
d,
H1(
a,
b,
c,
f,
g),
h) =
H1(
a,
c,
d,
f' * g',
h)
by A10, A12, ALTCAT_1:def 10
.=
(f' * g') * h'
by A14, A15, ALTCAT_1:def 10
.=
f' * (g' * h')
by A2, A10, A12, A14, ALTCAT_1:25
.=
H1(
a,
b,
d,
f,
g' * h')
by A10, A16, ALTCAT_1:def 10
.=
H1(
a,
b,
d,
f,
H1(
b,
c,
d,
g,
h))
by A12, A14, ALTCAT_1:def 10
;
verum end;
thus
B is associative
from YELLOW18:sch 2(A3, A8); verum