let S1 be OrderSortedSign; for U0 being non-empty OSAlgebra of S1 holds OSAlg_join U0 is commutative
let U0 be non-empty OSAlgebra of S1; OSAlg_join U0 is commutative
set o = OSAlg_join U0;
for x, y being Element of OSSub U0 holds (OSAlg_join U0) . (x,y) = (OSAlg_join U0) . (y,x)
proof
let x,
y be
Element of
OSSub U0;
(OSAlg_join U0) . (x,y) = (OSAlg_join U0) . (y,x)
reconsider U1 =
x,
U2 =
y as
strict OSSubAlgebra of
U0 by Def14;
set B = the
Sorts of
U1 (\/) the
Sorts of
U2;
the
Sorts of
U2 is
MSSubset of
U0
by MSUALG_2:def 9;
then A1:
the
Sorts of
U2 c= the
Sorts of
U0
by PBOOLE:def 18;
the
Sorts of
U1 is
MSSubset of
U0
by MSUALG_2:def 9;
then
the
Sorts of
U1 c= the
Sorts of
U0
by PBOOLE:def 18;
then
the
Sorts of
U1 (\/) the
Sorts of
U2 c= the
Sorts of
U0
by A1, PBOOLE:16;
then A2:
the
Sorts of
U1 (\/) the
Sorts of
U2 is
MSSubset of
U0
by PBOOLE:def 18;
( the
Sorts of
U1 is
OrderSortedSet of
S1 & the
Sorts of
U2 is
OrderSortedSet of
S1 )
by OSALG_1:17;
then
the
Sorts of
U1 (\/) the
Sorts of
U2 is
OrderSortedSet of
S1
by Th2;
then reconsider B = the
Sorts of
U1 (\/) the
Sorts of
U2 as
OSSubset of
U0 by A2, Def2;
A3:
U1 "\/"_os U2 = GenOSAlg B
by Def13;
(
(OSAlg_join U0) . (
x,
y)
= U1 "\/"_os U2 &
(OSAlg_join U0) . (
y,
x)
= U2 "\/"_os U1 )
by Def15;
hence
(OSAlg_join U0) . (
x,
y)
= (OSAlg_join U0) . (
y,
x)
by A3, Def13;
verum
end;
hence
OSAlg_join U0 is commutative
by BINOP_1:def 2; verum