:: ALGSTR_3 semantic presentation
:: deftheorem Def1 defines Tern ALGSTR_3:def 1 :
:: deftheorem Def2 ALGSTR_3:def 2 :
canceled;
:: deftheorem Def3 defines 1_ ALGSTR_3:def 3 :
definition
func ternaryreal -> TriOp of
REAL means :
Def4:
:: ALGSTR_3:def 4
for b
1, b
2, b
3 being
Real holds a
1 . b
1,b
2,b
3 = (b1 * b2) + b
3;
existence
ex b1 being TriOp of REAL st
for b2, b3, b4 being Real holds b1 . b2,b3,b4 = (b2 * b3) + b4
uniqueness
for b1, b2 being TriOp of REAL holds
( ( for b3, b4, b5 being Real holds b1 . b3,b4,b5 = (b3 * b4) + b5 ) & ( for b3, b4, b5 being Real holds b2 . b3,b4,b5 = (b3 * b4) + b5 ) implies b1 = b2 )
end;
:: deftheorem Def4 defines ternaryreal ALGSTR_3:def 4 :
:: deftheorem Def5 defines TernaryFieldEx ALGSTR_3:def 5 :
:: deftheorem Def6 defines tern ALGSTR_3:def 6 :
theorem Th1: :: ALGSTR_3:1
canceled;
theorem Th2: :: ALGSTR_3:2
canceled;
theorem Th3: :: ALGSTR_3:3
for b
1, b
2, b
3, b
4 being
Real holds
not ( b
1 <> b
2 & ( for b
5 being
Real holds
not
(b1 * b5) + b
3 = (b2 * b5) + b
4 ) )
theorem Th4: :: ALGSTR_3:4
canceled;
theorem Th5: :: ALGSTR_3:5
theorem Th6: :: ALGSTR_3:6
theorem Th7: :: ALGSTR_3:7
Lemma5:
for b1 being Scalar of TernaryFieldEx holds Tern b1,(1_ TernaryFieldEx ),(0. TernaryFieldEx ) = b1
Lemma6:
for b1 being Scalar of TernaryFieldEx holds Tern (1_ TernaryFieldEx ),b1,(0. TernaryFieldEx ) = b1
Lemma7:
for b1, b2 being Scalar of TernaryFieldEx holds Tern b1,(0. TernaryFieldEx ),b2 = b2
Lemma8:
for b1, b2 being Scalar of TernaryFieldEx holds Tern (0. TernaryFieldEx ),b1,b2 = b2
Lemma9:
for b1, b2, b3 being Scalar of TernaryFieldEx holds
ex b4 being Scalar of TernaryFieldEx st Tern b1,b2,b4 = b3
Lemma10:
for b1, b2, b3, b4 being Scalar of TernaryFieldEx holds
( Tern b1,b2,b3 = Tern b1,b2,b4 implies b3 = b4 )
Lemma11:
for b1, b2 being Scalar of TernaryFieldEx holds
( b1 <> b2 implies for b3, b4 being Scalar of TernaryFieldEx holds
ex b5, b6 being Scalar of TernaryFieldEx st
( Tern b5,b1,b6 = b3 & Tern b5,b2,b6 = b4 ) )
Lemma12:
for b1, b2 being Scalar of TernaryFieldEx holds
( b1 <> b2 implies for b3, b4 being Scalar of TernaryFieldEx holds
ex b5 being Scalar of TernaryFieldEx st Tern b1,b5,b3 = Tern b2,b5,b4 )
Lemma13:
for b1, b2, b3, b4, b5, b6 being Scalar of TernaryFieldEx holds
not ( Tern b3,b1,b5 = Tern b4,b1,b6 & Tern b3,b2,b5 = Tern b4,b2,b6 & not b1 = b2 & not b3 = b4 )
definition
let c
1 be non
empty TernaryFieldStr ;
attr a
1 is
Ternary-Field-like means :
Def7:
:: ALGSTR_3:def 7
(
0. a
1 <> 1_ a
1 & ( for b
1 being
Scalar of a
1 holds
Tern b
1,
(1_ a1),
(0. a1) = b
1 ) & ( for b
1 being
Scalar of a
1 holds
Tern (1_ a1),b
1,
(0. a1) = b
1 ) & ( for b
1, b
2 being
Scalar of a
1 holds
Tern b
1,
(0. a1),b
2 = b
2 ) & ( for b
1, b
2 being
Scalar of a
1 holds
Tern (0. a1),b
1,b
2 = b
2 ) & ( for b
1, b
2, b
3 being
Scalar of a
1 holds
ex b
4 being
Scalar of a
1 st
Tern b
1,b
2,b
4 = b
3 ) & ( for b
1, b
2, b
3, b
4 being
Scalar of a
1 holds
(
Tern b
1,b
2,b
3 = Tern b
1,b
2,b
4 implies b
3 = b
4 ) ) & ( for b
1, b
2 being
Scalar of a
1 holds
( b
1 <> b
2 implies for b
3, b
4 being
Scalar of a
1 holds
ex b
5, b
6 being
Scalar of a
1 st
(
Tern b
5,b
1,b
6 = b
3 &
Tern b
5,b
2,b
6 = b
4 ) ) ) & ( for b
1, b
2 being
Scalar of a
1 holds
( b
1 <> b
2 implies for b
3, b
4 being
Scalar of a
1 holds
ex b
5 being
Scalar of a
1 st
Tern b
1,b
5,b
3 = Tern b
2,b
5,b
4 ) ) & ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Scalar of a
1 holds
not (
Tern b
3,b
1,b
5 = Tern b
4,b
1,b
6 &
Tern b
3,b
2,b
5 = Tern b
4,b
2,b
6 & not b
1 = b
2 & not b
3 = b
4 ) ) );
end;
:: deftheorem Def7 defines Ternary-Field-like ALGSTR_3:def 7 :
for b
1 being non
empty TernaryFieldStr holds
( b
1 is
Ternary-Field-like iff (
0. b
1 <> 1_ b
1 & ( for b
2 being
Scalar of b
1 holds
Tern b
2,
(1_ b1),
(0. b1) = b
2 ) & ( for b
2 being
Scalar of b
1 holds
Tern (1_ b1),b
2,
(0. b1) = b
2 ) & ( for b
2, b
3 being
Scalar of b
1 holds
Tern b
2,
(0. b1),b
3 = b
3 ) & ( for b
2, b
3 being
Scalar of b
1 holds
Tern (0. b1),b
2,b
3 = b
3 ) & ( for b
2, b
3, b
4 being
Scalar of b
1 holds
ex b
5 being
Scalar of b
1 st
Tern b
2,b
3,b
5 = b
4 ) & ( for b
2, b
3, b
4, b
5 being
Scalar of b
1 holds
(
Tern b
2,b
3,b
4 = Tern b
2,b
3,b
5 implies b
4 = b
5 ) ) & ( for b
2, b
3 being
Scalar of b
1 holds
( b
2 <> b
3 implies for b
4, b
5 being
Scalar of b
1 holds
ex b
6, b
7 being
Scalar of b
1 st
(
Tern b
6,b
2,b
7 = b
4 &
Tern b
6,b
3,b
7 = b
5 ) ) ) & ( for b
2, b
3 being
Scalar of b
1 holds
( b
2 <> b
3 implies for b
4, b
5 being
Scalar of b
1 holds
ex b
6 being
Scalar of b
1 st
Tern b
2,b
6,b
4 = Tern b
3,b
6,b
5 ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Scalar of b
1 holds
not (
Tern b
4,b
2,b
6 = Tern b
5,b
2,b
7 &
Tern b
4,b
3,b
6 = Tern b
5,b
3,b
7 & not b
2 = b
3 & not b
4 = b
5 ) ) ) );
theorem Th8: :: ALGSTR_3:8
for b
1 being
Ternary-Fieldfor b
2, b
3, b
4, b
5, b
6, b
7 being
Scalar of b
1 holds
( b
2 <> b
3 &
Tern b
4,b
2,b
5 = Tern b
6,b
2,b
7 &
Tern b
4,b
3,b
5 = Tern b
6,b
3,b
7 implies ( b
4 = b
6 & b
5 = b
7 ) )
theorem Th9: :: ALGSTR_3:9
canceled;
theorem Th10: :: ALGSTR_3:10
canceled;
theorem Th11: :: ALGSTR_3:11
theorem Th12: :: ALGSTR_3:12
theorem Th13: :: ALGSTR_3:13
theorem Th14: :: ALGSTR_3:14