Article GROUP_7, MML version 4.99.1005
:: GROUP_7:th 1
theorem
for b1, b2 being set
st <*b1*> = <*b2*>
holds b1 = b2;
:: GROUP_7:th 2
theorem
for b1, b2, b3, b4 being set
st <*b1,b2*> = <*b3,b4*>
holds b1 = b3 & b2 = b4;
:: GROUP_7:th 3
theorem
for b1, b2, b3, b4, b5, b6 being set
st <*b1,b2,b3*> = <*b4,b5,b6*>
holds b1 = b4 & b2 = b5 & b3 = b6;
:: GROUP_7:attrnot 1 => GROUP_7:attr 1
definition
let a1 be Relation-like set;
attr a1 is HGrStr-yielding means
for b1 being set
st b1 in proj2 a1
holds b1 is non empty multMagma;
end;
:: GROUP_7:dfs 1
definiens
let a1 be Relation-like set;
To prove
a1 is HGrStr-yielding
it is sufficient to prove
thus for b1 being set
st b1 in proj2 a1
holds b1 is non empty multMagma;
:: GROUP_7:def 1
theorem
for b1 being Relation-like set holds
b1 is HGrStr-yielding
iff
for b2 being set
st b2 in proj2 b1
holds b2 is non empty multMagma;
:: GROUP_7:condreg 1
registration
cluster Relation-like Function-like HGrStr-yielding -> 1-sorted-yielding (set);
end;
:: GROUP_7:exreg 1
registration
let a1 be set;
cluster Relation-like Function-like HGrStr-yielding ManySortedSet of a1;
end;
:: GROUP_7:exreg 2
registration
cluster Relation-like Function-like HGrStr-yielding set;
end;
:: GROUP_7:modenot 1
definition
let a1 be set;
mode HGrStr-Family of a1 is HGrStr-yielding ManySortedSet of a1;
end;
:: GROUP_7:funcnot 1 => GROUP_7:func 1
definition
let a1 be non empty set;
let a2 be HGrStr-yielding ManySortedSet of a1;
let a3 be Element of a1;
redefine func a2 . a3 -> non empty multMagma;
end;
:: GROUP_7:funcreg 1
registration
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
cluster Carrier a2 -> non-empty;
end;
:: GROUP_7:funcnot 2 => GROUP_7:func 2
definition
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
func product A2 -> strict multMagma means
the carrier of it = product Carrier a2 &
(for b1, b2 being Element of product Carrier a2
for b3 being set
st b3 in a1
holds ex b4 being non empty multMagma st
ex b5 being Relation-like Function-like set st
b4 = a2 . b3 &
b5 = (the multF of it) .(b1,b2) &
b5 . b3 = (the multF of b4) .(b1 . b3,b2 . b3));
end;
:: GROUP_7:def 2
theorem
for b1 being set
for b2 being HGrStr-yielding ManySortedSet of b1
for b3 being strict multMagma holds
b3 = product b2
iff
the carrier of b3 = product Carrier b2 &
(for b4, b5 being Element of product Carrier b2
for b6 being set
st b6 in b1
holds ex b7 being non empty multMagma st
ex b8 being Relation-like Function-like set st
b7 = b2 . b6 &
b8 = (the multF of b3) .(b4,b5) &
b8 . b6 = (the multF of b7) .(b4 . b6,b5 . b6));
:: GROUP_7:funcreg 2
registration
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
cluster product a2 -> non empty strict constituted-Functions;
end;
:: GROUP_7:th 4
theorem
for b1, b2 being set
for b3, b4, b5 being Relation-like Function-like set
for b6 being HGrStr-yielding ManySortedSet of b1
for b7 being non empty multMagma
for b8, b9 being Element of the carrier of product b6
for b10, b11 being Element of the carrier of b7
st b2 in b1 & b7 = b6 . b2 & b3 = b8 & b4 = b9 & b5 = b8 * b9 & b3 . b2 = b10 & b4 . b2 = b11
holds b10 * b11 = b5 . b2;
:: GROUP_7:attrnot 2 => GROUP_7:attr 2
definition
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
attr a2 is Group-like means
for b1 being set
st b1 in a1
holds ex b2 being non empty Group-like multMagma st
b2 = a2 . b1;
end;
:: GROUP_7:dfs 3
definiens
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
To prove
a2 is Group-like
it is sufficient to prove
thus for b1 being set
st b1 in a1
holds ex b2 being non empty Group-like multMagma st
b2 = a2 . b1;
:: GROUP_7:def 3
theorem
for b1 being set
for b2 being HGrStr-yielding ManySortedSet of b1 holds
b2 is Group-like(b1)
iff
for b3 being set
st b3 in b1
holds ex b4 being non empty Group-like multMagma st
b4 = b2 . b3;
:: GROUP_7:attrnot 3 => GROUP_7:attr 3
definition
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
attr a2 is associative means
for b1 being set
st b1 in a1
holds ex b2 being non empty associative multMagma st
b2 = a2 . b1;
end;
:: GROUP_7:dfs 4
definiens
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
To prove
a2 is associative
it is sufficient to prove
thus for b1 being set
st b1 in a1
holds ex b2 being non empty associative multMagma st
b2 = a2 . b1;
:: GROUP_7:def 4
theorem
for b1 being set
for b2 being HGrStr-yielding ManySortedSet of b1 holds
b2 is associative(b1)
iff
for b3 being set
st b3 in b1
holds ex b4 being non empty associative multMagma st
b4 = b2 . b3;
:: GROUP_7:attrnot 4 => GROUP_7:attr 4
definition
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
attr a2 is commutative means
for b1 being set
st b1 in a1
holds ex b2 being non empty commutative multMagma st
b2 = a2 . b1;
end;
:: GROUP_7:dfs 5
definiens
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
To prove
a2 is commutative
it is sufficient to prove
thus for b1 being set
st b1 in a1
holds ex b2 being non empty commutative multMagma st
b2 = a2 . b1;
:: GROUP_7:def 5
theorem
for b1 being set
for b2 being HGrStr-yielding ManySortedSet of b1 holds
b2 is commutative(b1)
iff
for b3 being set
st b3 in b1
holds ex b4 being non empty commutative multMagma st
b4 = b2 . b3;
:: GROUP_7:attrnot 5 => GROUP_7:attr 2
definition
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
attr a2 is Group-like means
for b1 being Element of a1 holds
a2 . b1 is Group-like;
end;
:: GROUP_7:dfs 6
definiens
let a1 be non empty set;
let a2 be HGrStr-yielding ManySortedSet of a1;
To prove
a2 is Group-like
it is sufficient to prove
thus for b1 being Element of a1 holds
a2 . b1 is Group-like;
:: GROUP_7:def 6
theorem
for b1 being non empty set
for b2 being HGrStr-yielding ManySortedSet of b1 holds
b2 is Group-like(b1)
iff
for b3 being Element of b1 holds
b2 . b3 is Group-like;
:: GROUP_7:attrnot 6 => GROUP_7:attr 3
definition
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
attr a2 is associative means
for b1 being Element of a1 holds
a2 . b1 is associative;
end;
:: GROUP_7:dfs 7
definiens
let a1 be non empty set;
let a2 be HGrStr-yielding ManySortedSet of a1;
To prove
a2 is associative
it is sufficient to prove
thus for b1 being Element of a1 holds
a2 . b1 is associative;
:: GROUP_7:def 7
theorem
for b1 being non empty set
for b2 being HGrStr-yielding ManySortedSet of b1 holds
b2 is associative(b1)
iff
for b3 being Element of b1 holds
b2 . b3 is associative;
:: GROUP_7:attrnot 7 => GROUP_7:attr 4
definition
let a1 be set;
let a2 be HGrStr-yielding ManySortedSet of a1;
attr a2 is commutative means
for b1 being Element of a1 holds
a2 . b1 is commutative;
end;
:: GROUP_7:dfs 8
definiens
let a1 be non empty set;
let a2 be HGrStr-yielding ManySortedSet of a1;
To prove
a2 is commutative
it is sufficient to prove
thus for b1 being Element of a1 holds
a2 . b1 is commutative;
:: GROUP_7:def 8
theorem
for b1 being non empty set
for b2 being HGrStr-yielding ManySortedSet of b1 holds
b2 is commutative(b1)
iff
for b3 being Element of b1 holds
b2 . b3 is commutative;
:: GROUP_7:exreg 3
registration
let a1 be set;
cluster Relation-like Function-like 1-sorted-yielding HGrStr-yielding Group-like associative commutative ManySortedSet of a1;
end;
:: GROUP_7:funcreg 3
registration
let a1 be set;
let a2 be HGrStr-yielding Group-like ManySortedSet of a1;
cluster product a2 -> strict Group-like;
end;
:: GROUP_7:funcreg 4
registration
let a1 be set;
let a2 be HGrStr-yielding associative ManySortedSet of a1;
cluster product a2 -> strict associative;
end;
:: GROUP_7:funcreg 5
registration
let a1 be set;
let a2 be HGrStr-yielding commutative ManySortedSet of a1;
cluster product a2 -> strict commutative;
end;
:: GROUP_7:th 5
theorem
for b1, b2 being set
for b3 being HGrStr-yielding ManySortedSet of b1
for b4 being non empty multMagma
st b2 in b1 & b4 = b3 . b2 & product b3 is Group-like
holds b4 is Group-like;
:: GROUP_7:th 6
theorem
for b1, b2 being set
for b3 being HGrStr-yielding ManySortedSet of b1
for b4 being non empty multMagma
st b2 in b1 & b4 = b3 . b2 & product b3 is associative
holds b4 is associative;
:: GROUP_7:th 7
theorem
for b1, b2 being set
for b3 being HGrStr-yielding ManySortedSet of b1
for b4 being non empty multMagma
st b2 in b1 & b4 = b3 . b2 & product b3 is commutative
holds b4 is commutative;
:: GROUP_7:th 8
theorem
for b1 being set
for b2 being ManySortedSet of b1
for b3 being HGrStr-yielding Group-like ManySortedSet of b1
st for b4 being set
st b4 in b1
holds ex b5 being non empty Group-like multMagma st
b5 = b3 . b4 & b2 . b4 = 1_ b5
holds b2 = 1_ product b3;
:: GROUP_7:th 9
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set
for b4 being HGrStr-yielding Group-like ManySortedSet of b1
for b5 being non empty Group-like multMagma
st b2 in b1 & b5 = b4 . b2 & b3 = 1_ product b4
holds b3 . b2 = 1_ b5;
:: GROUP_7:th 10
theorem
for b1 being set
for b2 being Relation-like Function-like set
for b3 being ManySortedSet of b1
for b4 being HGrStr-yielding Group-like associative ManySortedSet of b1
for b5 being Element of the carrier of product b4
st b5 = b2 &
(for b6 being set
st b6 in b1
holds ex b7 being non empty Group-like associative multMagma st
ex b8 being Element of the carrier of b7 st
b7 = b4 . b6 & b3 . b6 = b8 " & b8 = b2 . b6)
holds b3 = b5 ";
:: GROUP_7:th 11
theorem
for b1, b2 being set
for b3, b4 being Relation-like Function-like set
for b5 being HGrStr-yielding Group-like associative ManySortedSet of b1
for b6 being Element of the carrier of product b5
for b7 being non empty Group-like associative multMagma
for b8 being Element of the carrier of b7
st b2 in b1 & b7 = b5 . b2 & b3 = b6 & b4 = b6 " & b3 . b2 = b8
holds b4 . b2 = b8 ";
:: GROUP_7:funcnot 3 => GROUP_7:func 3
definition
let a1 be set;
let a2 be HGrStr-yielding Group-like associative ManySortedSet of a1;
func sum A2 -> strict Subgroup of product a2 means
for b1 being set holds
b1 in the carrier of it
iff
ex b2 being Element of product Carrier a2 st
ex b3 being finite Element of bool a1 st
ex b4 being ManySortedSet of b3 st
b2 = 1_ product a2 &
b1 = b2 +* b4 &
(for b5 being set
st b5 in b3
holds ex b6 being non empty Group-like multMagma st
b6 = a2 . b5 & b4 . b5 in the carrier of b6 & b4 . b5 <> 1_ b6);
end;
:: GROUP_7:def 9
theorem
for b1 being set
for b2 being HGrStr-yielding Group-like associative ManySortedSet of b1
for b3 being strict Subgroup of product b2 holds
b3 = sum b2
iff
for b4 being set holds
b4 in the carrier of b3
iff
ex b5 being Element of product Carrier b2 st
ex b6 being finite Element of bool b1 st
ex b7 being ManySortedSet of b6 st
b5 = 1_ product b2 &
b4 = b5 +* b7 &
(for b8 being set
st b8 in b6
holds ex b9 being non empty Group-like multMagma st
b9 = b2 . b8 & b7 . b8 in the carrier of b9 & b7 . b8 <> 1_ b9);
:: GROUP_7:funcreg 6
registration
let a1 be set;
let a2 be HGrStr-yielding Group-like associative ManySortedSet of a1;
let a3, a4 be Element of the carrier of sum a2;
cluster (the multF of sum a2) .(a3,a4) -> Relation-like Function-like;
end;
:: GROUP_7:th 12
theorem
for b1 being finite set
for b2 being HGrStr-yielding Group-like associative ManySortedSet of b1 holds
product b2 = sum b2;
:: GROUP_7:th 13
theorem
for b1 being non empty multMagma holds
<*b1*> is HGrStr-yielding ManySortedSet of {1};
:: GROUP_7:funcnot 4 => GROUP_7:func 4
definition
let a1 be non empty multMagma;
redefine func <*a1*> -> HGrStr-yielding ManySortedSet of {1};
end;
:: GROUP_7:th 14
theorem
for b1 being non empty Group-like multMagma holds
<*b1*> is HGrStr-yielding Group-like ManySortedSet of {1};
:: GROUP_7:funcnot 5 => GROUP_7:func 5
definition
let a1 be non empty Group-like multMagma;
redefine func <*a1*> -> HGrStr-yielding Group-like ManySortedSet of {1};
end;
:: GROUP_7:th 15
theorem
for b1 being non empty associative multMagma holds
<*b1*> is HGrStr-yielding associative ManySortedSet of {1};
:: GROUP_7:funcnot 6 => GROUP_7:func 6
definition
let a1 be non empty associative multMagma;
redefine func <*a1*> -> HGrStr-yielding associative ManySortedSet of {1};
end;
:: GROUP_7:th 16
theorem
for b1 being non empty commutative multMagma holds
<*b1*> is HGrStr-yielding commutative ManySortedSet of {1};
:: GROUP_7:funcnot 7 => GROUP_7:func 7
definition
let a1 be non empty commutative multMagma;
redefine func <*a1*> -> HGrStr-yielding commutative ManySortedSet of {1};
end;
:: GROUP_7:th 17
theorem
for b1 being non empty Group-like associative multMagma holds
<*b1*> is HGrStr-yielding Group-like associative ManySortedSet of {1};
:: GROUP_7:funcnot 8 => GROUP_7:func 8
definition
let a1 be non empty Group-like associative multMagma;
redefine func <*a1*> -> HGrStr-yielding Group-like associative ManySortedSet of {1};
end;
:: GROUP_7:th 18
theorem
for b1 being non empty Group-like associative commutative multMagma holds
<*b1*> is HGrStr-yielding Group-like associative commutative ManySortedSet of {1};
:: GROUP_7:funcnot 9 => GROUP_7:func 9
definition
let a1 be non empty Group-like associative commutative multMagma;
redefine func <*a1*> -> HGrStr-yielding Group-like associative commutative ManySortedSet of {1};
end;
:: GROUP_7:condreg 2
registration
let a1 be non empty multMagma;
cluster -> FinSequence-like (Element of product Carrier <*a1*>);
end;
:: GROUP_7:condreg 3
registration
let a1 be non empty multMagma;
cluster -> FinSequence-like (Element of the carrier of product <*a1*>);
end;
:: GROUP_7:funcnot 10 => GROUP_7:func 10
definition
let a1 be non empty multMagma;
let a2 be Element of the carrier of a1;
redefine func <*a2*> -> Element of the carrier of product <*a1*>;
end;
:: GROUP_7:th 19
theorem
for b1, b2 being non empty multMagma holds
<*b1,b2*> is HGrStr-yielding ManySortedSet of {1,2};
:: GROUP_7:funcnot 11 => GROUP_7:func 11
definition
let a1, a2 be non empty multMagma;
redefine func <*a1, a2*> -> HGrStr-yielding ManySortedSet of {1,2};
end;
:: GROUP_7:th 20
theorem
for b1, b2 being non empty Group-like multMagma holds
<*b1,b2*> is HGrStr-yielding Group-like ManySortedSet of {1,2};
:: GROUP_7:funcnot 12 => GROUP_7:func 12
definition
let a1, a2 be non empty Group-like multMagma;
redefine func <*a1, a2*> -> HGrStr-yielding Group-like ManySortedSet of {1,2};
end;
:: GROUP_7:th 21
theorem
for b1, b2 being non empty associative multMagma holds
<*b1,b2*> is HGrStr-yielding associative ManySortedSet of {1,2};
:: GROUP_7:funcnot 13 => GROUP_7:func 13
definition
let a1, a2 be non empty associative multMagma;
redefine func <*a1, a2*> -> HGrStr-yielding associative ManySortedSet of {1,2};
end;
:: GROUP_7:th 22
theorem
for b1, b2 being non empty commutative multMagma holds
<*b1,b2*> is HGrStr-yielding commutative ManySortedSet of {1,2};
:: GROUP_7:funcnot 14 => GROUP_7:func 14
definition
let a1, a2 be non empty commutative multMagma;
redefine func <*a1, a2*> -> HGrStr-yielding commutative ManySortedSet of {1,2};
end;
:: GROUP_7:th 23
theorem
for b1, b2 being non empty Group-like associative multMagma holds
<*b1,b2*> is HGrStr-yielding Group-like associative ManySortedSet of {1,2};
:: GROUP_7:funcnot 15 => GROUP_7:func 15
definition
let a1, a2 be non empty Group-like associative multMagma;
redefine func <*a1, a2*> -> HGrStr-yielding Group-like associative ManySortedSet of {1,2};
end;
:: GROUP_7:th 24
theorem
for b1, b2 being non empty Group-like associative commutative multMagma holds
<*b1,b2*> is HGrStr-yielding Group-like associative commutative ManySortedSet of {1,2};
:: GROUP_7:funcnot 16 => GROUP_7:func 16
definition
let a1, a2 be non empty Group-like associative commutative multMagma;
redefine func <*a1, a2*> -> HGrStr-yielding Group-like associative commutative ManySortedSet of {1,2};
end;
:: GROUP_7:condreg 4
registration
let a1, a2 be non empty multMagma;
cluster -> FinSequence-like (Element of product Carrier <*a1,a2*>);
end;
:: GROUP_7:condreg 5
registration
let a1, a2 be non empty multMagma;
cluster -> FinSequence-like (Element of the carrier of product <*a1,a2*>);
end;
:: GROUP_7:funcnot 17 => GROUP_7:func 17
definition
let a1, a2 be non empty multMagma;
let a3 be Element of the carrier of a1;
let a4 be Element of the carrier of a2;
redefine func <*a3, a4*> -> Element of the carrier of product <*a1,a2*>;
end;
:: GROUP_7:th 25
theorem
for b1, b2, b3 being non empty multMagma holds
<*b1,b2,b3*> is HGrStr-yielding ManySortedSet of {1,2,3};
:: GROUP_7:funcnot 18 => GROUP_7:func 18
definition
let a1, a2, a3 be non empty multMagma;
redefine func <*a1, a2, a3*> -> HGrStr-yielding ManySortedSet of {1,2,3};
end;
:: GROUP_7:th 26
theorem
for b1, b2, b3 being non empty Group-like multMagma holds
<*b1,b2,b3*> is HGrStr-yielding Group-like ManySortedSet of {1,2,3};
:: GROUP_7:funcnot 19 => GROUP_7:func 19
definition
let a1, a2, a3 be non empty Group-like multMagma;
redefine func <*a1, a2, a3*> -> HGrStr-yielding Group-like ManySortedSet of {1,2,3};
end;
:: GROUP_7:th 27
theorem
for b1, b2, b3 being non empty associative multMagma holds
<*b1,b2,b3*> is HGrStr-yielding associative ManySortedSet of {1,2,3};
:: GROUP_7:funcnot 20 => GROUP_7:func 20
definition
let a1, a2, a3 be non empty associative multMagma;
redefine func <*a1, a2, a3*> -> HGrStr-yielding associative ManySortedSet of {1,2,3};
end;
:: GROUP_7:th 28
theorem
for b1, b2, b3 being non empty commutative multMagma holds
<*b1,b2,b3*> is HGrStr-yielding commutative ManySortedSet of {1,2,3};
:: GROUP_7:funcnot 21 => GROUP_7:func 21
definition
let a1, a2, a3 be non empty commutative multMagma;
redefine func <*a1, a2, a3*> -> HGrStr-yielding commutative ManySortedSet of {1,2,3};
end;
:: GROUP_7:th 29
theorem
for b1, b2, b3 being non empty Group-like associative multMagma holds
<*b1,b2,b3*> is HGrStr-yielding Group-like associative ManySortedSet of {1,2,3};
:: GROUP_7:funcnot 22 => GROUP_7:func 22
definition
let a1, a2, a3 be non empty Group-like associative multMagma;
redefine func <*a1, a2, a3*> -> HGrStr-yielding Group-like associative ManySortedSet of {1,2,3};
end;
:: GROUP_7:th 30
theorem
for b1, b2, b3 being non empty Group-like associative commutative multMagma holds
<*b1,b2,b3*> is HGrStr-yielding Group-like associative commutative ManySortedSet of {1,2,3};
:: GROUP_7:funcnot 23 => GROUP_7:func 23
definition
let a1, a2, a3 be non empty Group-like associative commutative multMagma;
redefine func <*a1, a2, a3*> -> HGrStr-yielding Group-like associative commutative ManySortedSet of {1,2,3};
end;
:: GROUP_7:condreg 6
registration
let a1, a2, a3 be non empty multMagma;
cluster -> FinSequence-like (Element of product Carrier <*a1,a2,a3*>);
end;
:: GROUP_7:condreg 7
registration
let a1, a2, a3 be non empty multMagma;
cluster -> FinSequence-like (Element of the carrier of product <*a1,a2,a3*>);
end;
:: GROUP_7:funcnot 24 => GROUP_7:func 24
definition
let a1, a2, a3 be non empty multMagma;
let a4 be Element of the carrier of a1;
let a5 be Element of the carrier of a2;
let a6 be Element of the carrier of a3;
redefine func <*a4, a5, a6*> -> Element of the carrier of product <*a1,a2,a3*>;
end;
:: GROUP_7:th 31
theorem
for b1 being non empty multMagma
for b2, b3 being Element of the carrier of b1 holds
<*b2*> * <*b3*> = <*b2 * b3*>;
:: GROUP_7:th 32
theorem
for b1, b2 being non empty multMagma
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2 holds
<*b3,b5*> * <*b4,b6*> = <*b3 * b4,b5 * b6*>;
:: GROUP_7:th 33
theorem
for b1, b2, b3 being non empty multMagma
for b4, b5 being Element of the carrier of b1
for b6, b7 being Element of the carrier of b2
for b8, b9 being Element of the carrier of b3 holds
<*b4,b6,b8*> * <*b5,b7,b9*> = <*b4 * b5,b6 * b7,b8 * b9*>;
:: GROUP_7:th 34
theorem
for b1 being non empty Group-like multMagma holds
1_ product <*b1*> = <*1_ b1*>;
:: GROUP_7:th 35
theorem
for b1, b2 being non empty Group-like multMagma holds
1_ product <*b1,b2*> = <*1_ b1,1_ b2*>;
:: GROUP_7:th 36
theorem
for b1, b2, b3 being non empty Group-like multMagma holds
1_ product <*b1,b2,b3*> = <*1_ b1,1_ b2,1_ b3*>;
:: GROUP_7:th 37
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
<*b2*> " = <*b2 "*>;
:: GROUP_7:th 38
theorem
for b1, b2 being non empty Group-like associative multMagma
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2 holds
<*b3,b4*> " = <*b3 ",b4 "*>;
:: GROUP_7:th 39
theorem
for b1, b2, b3 being non empty Group-like associative multMagma
for b4 being Element of the carrier of b1
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b3 holds
<*b4,b5,b6*> " = <*b4 ",b5 ",b6 "*>;
:: GROUP_7:th 40
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of product <*b1*>
st for b3 being Element of the carrier of b1 holds
b2 . b3 = <*b3*>
holds b2 is Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of product <*b1*>;
:: GROUP_7:th 41
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of product <*b1*>
st for b3 being Element of the carrier of b1 holds
b2 . b3 = <*b3*>
holds b2 is being_isomorphism(b1, product <*b1*>);
:: GROUP_7:th 42
theorem
for b1 being non empty Group-like associative multMagma holds
b1,product <*b1*> are_isomorphic;