let S be non empty non void ManySortedSign ; for U0 being non-empty MSAlgebra over S holds MSAlg_join U0 is commutative
let U0 be non-empty MSAlgebra over S; MSAlg_join U0 is commutative
set o = MSAlg_join U0;
for x, y being Element of MSSub U0 holds (MSAlg_join U0) . (x,y) = (MSAlg_join U0) . (y,x)
proof
let x,
y be
Element of
MSSub U0;
(MSAlg_join U0) . (x,y) = (MSAlg_join U0) . (y,x)
reconsider U1 =
x,
U2 =
y as
strict MSSubAlgebra of
U0 by Def19;
set B = the
Sorts of
U1 (\/) the
Sorts of
U2;
the
Sorts of
U2 is
MSSubset of
U0
by Def9;
then A1:
the
Sorts of
U2 c= the
Sorts of
U0
by PBOOLE:def 18;
the
Sorts of
U1 is
MSSubset of
U0
by Def9;
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 reconsider B = the
Sorts of
U1 (\/) the
Sorts of
U2 as
MSSubset of
U0 by PBOOLE:def 18;
A2:
U1 "\/" U2 = GenMSAlg B
by Def18;
(
(MSAlg_join U0) . (
x,
y)
= U1 "\/" U2 &
(MSAlg_join U0) . (
y,
x)
= U2 "\/" U1 )
by Def20;
hence
(MSAlg_join U0) . (
x,
y)
= (MSAlg_join U0) . (
y,
x)
by A2, Def18;
verum
end;
hence
MSAlg_join U0 is commutative
by BINOP_1:def 2; verum