:: MULTOP_1 semantic presentation
:: deftheorem Def1 defines . MULTOP_1:def 1 :
for b
1 being
Functionfor b
2, b
3, b
4 being
set holds b
1 . b
2,b
3,b
4 = b
1 . [b2,b3,b4];
definition
let c
1, c
2, c
3, c
4 be non
empty set ;
let c
5 be
Function of
[:c1,c2,c3:],c
4;
let c
6 be
Element of c
1;
let c
7 be
Element of c
2;
let c
8 be
Element of c
3;
redefine func . as c
5 . c
6,c
7,c
8 -> Element of a
4;
coherence
c5 . c6,c7,c8 is Element of c4
end;
theorem Th1: :: MULTOP_1:1
canceled;
theorem Th2: :: MULTOP_1:2
for b
1 being non
empty set for b
2, b
3, b
4 being
set for b
5, b
6 being
Function of
[:b2,b3,b4:],b
1 holds
( ( for b
7, b
8, b
9 being
set holds
( b
7 in b
2 & b
8 in b
3 & b
9 in b
4 implies b
5 . [b7,b8,b9] = b
6 . [b7,b8,b9] ) ) implies b
5 = b
6 )
theorem Th3: :: MULTOP_1:3
for b
1, b
2, b
3, b
4 being non
empty set for b
5, b
6 being
Function of
[:b1,b2,b3:],b
4 holds
( ( for b
7 being
Element of b
1for b
8 being
Element of b
2for b
9 being
Element of b
3 holds b
5 . [b7,b8,b9] = b
6 . [b7,b8,b9] ) implies b
5 = b
6 )
theorem Th4: :: MULTOP_1:4
for b
1, b
2, b
3, b
4 being non
empty set for b
5, b
6 being
Function of
[:b1,b2,b3:],b
4 holds
( ( for b
7 being
Element of b
1for b
8 being
Element of b
2for b
9 being
Element of b
3 holds b
5 . b
7,b
8,b
9 = b
6 . b
7,b
8,b
9 ) implies b
5 = b
6 )
scheme :: MULTOP_1:sch 1
s1{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , F
4()
-> non
empty set , P
1[
set ,
set ,
set ,
set ] } :
provided
E3:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2()
for b
3 being
Element of F
3() holds
ex b
4 being
Element of F
4() st P
1[b
1,b
2,b
3,b
4]
scheme :: MULTOP_1:sch 2
s2{ F
1()
-> non
empty set , P
1[
Element of F
1(),
Element of F
1(),
Element of F
1(),
Element of F
1()] } :
ex b
1 being
TriOp of F
1() st
for b
2, b
3, b
4 being
Element of F
1() holds P
1[b
2,b
3,b
4,b
1 . b
2,b
3,b
4]
provided
E3:
for b
1, b
2, b
3 being
Element of F
1() holds
ex b
4 being
Element of F
1() st P
1[b
1,b
2,b
3,b
4]
definition
let c
1 be
Function;
let c
2, c
3, c
4, c
5 be
set ;
func c
1 . c
2,c
3,c
4,c
5 -> set equals :: MULTOP_1:def 2
a
1 . [a2,a3,a4,a5];
correctness
coherence
c1 . [c2,c3,c4,c5] is set ;
;
end;
:: deftheorem Def2 defines . MULTOP_1:def 2 :
for b
1 being
Functionfor b
2, b
3, b
4, b
5 being
set holds b
1 . b
2,b
3,b
4,b
5 = b
1 . [b2,b3,b4,b5];
definition
let c
1, c
2, c
3, c
4, c
5 be non
empty set ;
let c
6 be
Function of
[:c1,c2,c3,c4:],c
5;
let c
7 be
Element of c
1;
let c
8 be
Element of c
2;
let c
9 be
Element of c
3;
let c
10 be
Element of c
4;
redefine func . as c
6 . c
7,c
8,c
9,c
10 -> Element of a
5;
coherence
c6 . c7,c8,c9,c10 is Element of c5
end;
theorem Th5: :: MULTOP_1:5
canceled;
theorem Th6: :: MULTOP_1:6
for b
1 being non
empty set for b
2, b
3, b
4, b
5 being
set for b
6, b
7 being
Function of
[:b2,b3,b4,b5:],b
1 holds
( ( for b
8, b
9, b
10, b
11 being
set holds
( b
8 in b
2 & b
9 in b
3 & b
10 in b
4 & b
11 in b
5 implies b
6 . [b8,b9,b10,b11] = b
7 . [b8,b9,b10,b11] ) ) implies b
6 = b
7 )
theorem Th7: :: MULTOP_1:7
for b
1, b
2, b
3, b
4, b
5 being non
empty set for b
6, b
7 being
Function of
[:b1,b2,b3,b4:],b
5 holds
( ( for b
8 being
Element of b
1for b
9 being
Element of b
2for b
10 being
Element of b
3for b
11 being
Element of b
4 holds b
6 . [b8,b9,b10,b11] = b
7 . [b8,b9,b10,b11] ) implies b
6 = b
7 )
theorem Th8: :: MULTOP_1:8
for b
1, b
2, b
3, b
4, b
5 being non
empty set for b
6, b
7 being
Function of
[:b1,b2,b3,b4:],b
5 holds
( ( for b
8 being
Element of b
1for b
9 being
Element of b
2for b
10 being
Element of b
3for b
11 being
Element of b
4 holds b
6 . b
8,b
9,b
10,b
11 = b
7 . b
8,b
9,b
10,b
11 ) implies b
6 = b
7 )
scheme :: MULTOP_1:sch 5
s5{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , F
4()
-> non
empty set , F
5()
-> non
empty set , P
1[
set ,
set ,
set ,
set ,
set ] } :
ex b
1 being
Function of
[:F1(),F2(),F3(),F4():],F
5() st
for b
2 being
Element of F
1()
for b
3 being
Element of F
2()
for b
4 being
Element of F
3()
for b
5 being
Element of F
4() holds P
1[b
2,b
3,b
4,b
5,b
1 . [b2,b3,b4,b5]]
provided
scheme :: MULTOP_1:sch 6
s6{ F
1()
-> non
empty set , P
1[
Element of F
1(),
Element of F
1(),
Element of F
1(),
Element of F
1(),
Element of F
1()] } :
ex b
1 being
QuaOp of F
1() st
for b
2, b
3, b
4, b
5 being
Element of F
1() holds P
1[b
2,b
3,b
4,b
5,b
1 . b
2,b
3,b
4,b
5]
provided
E5:
for b
1, b
2, b
3, b
4 being
Element of F
1() holds
ex b
5 being
Element of F
1() st P
1[b
1,b
2,b
3,b
4,b
5]
scheme :: MULTOP_1:sch 7
s7{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , F
4()
-> non
empty set , F
5()
-> non
empty set , F
6(
Element of F
1(),
Element of F
2(),
Element of F
3(),
Element of F
4())
-> Element of F
5() } :
ex b
1 being
Function of
[:F1(),F2(),F3(),F4():],F
5() st
for b
2 being
Element of F
1()
for b
3 being
Element of F
2()
for b
4 being
Element of F
3()
for b
5 being
Element of F
4() holds b
1 . [b2,b3,b4,b5] = F
6(b
2,b
3,b
4,b
5)
scheme :: MULTOP_1:sch 8
s8{ F
1()
-> non
empty set , F
2(
Element of F
1(),
Element of F
1(),
Element of F
1(),
Element of F
1())
-> Element of F
1() } :
ex b
1 being
QuaOp of F
1() st
for b
2, b
3, b
4, b
5 being
Element of F
1() holds b
1 . b
2,b
3,b
4,b
5 = F
2(b
2,b
3,b
4,b
5)