Article SETWOP_2, MML version 4.99.1005

:: SETWOP_2:th 3
theorem
for b1, b2 being non empty set
for b3, b4 being Element of b1
for b5 being Function-like quasi_total Relation of [:b2,b2:],b2
for b6 being Function-like quasi_total Relation of b1,b2
      st b5 is commutative(b2) & b5 is associative(b2) & b3 <> b4
   holds b5 $$({.b3,b4.},b6) = b5 .(b6 . b3,b6 . b4);

:: SETWOP_2:th 4
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being Element of Fin b1
for b5 being Function-like quasi_total Relation of [:b2,b2:],b2
for b6 being Function-like quasi_total Relation of b1,b2
      st b5 is commutative(b2) & b5 is associative(b2) & (b4 = {} implies b5 is having_a_unity(b2)) & not b3 in b4
   holds b5 $$(b4 \/ {.b3.},b6) = b5 .(b5 $$(b4,b6),b6 . b3);

:: SETWOP_2:th 5
theorem
for b1, b2 being non empty set
for b3, b4, b5 being Element of b1
for b6 being Function-like quasi_total Relation of [:b2,b2:],b2
for b7 being Function-like quasi_total Relation of b1,b2
      st b6 is commutative(b2) & b6 is associative(b2) & b3 <> b4 & b3 <> b5 & b4 <> b5
   holds b6 $$({.b3,b4,b5.},b7) = b6 .(b6 .(b7 . b3,b7 . b4),b7 . b5);

:: SETWOP_2:th 6
theorem
for b1, b2 being non empty set
for b3, b4 being Element of Fin b1
for b5 being Function-like quasi_total Relation of [:b2,b2:],b2
for b6 being Function-like quasi_total Relation of b1,b2
      st b5 is commutative(b2) &
         b5 is associative(b2) &
         (b3 <> {} & b4 <> {} or b5 is having_a_unity(b2)) &
         b3 misses b4
   holds b5 $$(b3 \/ b4,b6) = b5 .(b5 $$(b3,b6),b5 $$(b4,b6));

:: SETWOP_2:th 7
theorem
for b1, b2, b3 being non empty set
for b4 being Element of Fin b1
for b5 being Element of Fin b2
for b6 being Function-like quasi_total Relation of [:b3,b3:],b3
for b7 being Function-like quasi_total Relation of b1,b3
for b8 being Function-like quasi_total Relation of b2,b3
      st b6 is commutative(b3) &
         b6 is associative(b3) &
         (b5 = {} implies b6 is having_a_unity(b3)) &
         (ex b9 being Relation-like Function-like set st
            proj1 b9 = b5 & proj2 b9 = b4 & b9 is one-to-one & b8 | b5 = b9 * b7)
   holds b6 $$(b5,b8) = b6 $$(b4,b7);

:: SETWOP_2:th 8
theorem
for b1, b2, b3 being non empty set
for b4 being Element of Fin b1
for b5 being Function-like quasi_total Relation of b1,b2
for b6 being Function-like quasi_total Relation of [:b3,b3:],b3
for b7 being Function-like quasi_total Relation of b2,b3
      st b6 is commutative(b3) & b6 is associative(b3) & (b4 = {} implies b6 is having_a_unity(b3)) & b5 is one-to-one
   holds b6 $$(b5 .: b4,b7) = b6 $$(b4,b7 * b5);

:: SETWOP_2:th 9
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4 being Function-like quasi_total Relation of [:b2,b2:],b2
for b5, b6 being Function-like quasi_total Relation of b1,b2
      st b4 is commutative(b2) & b4 is associative(b2) & (b3 = {} implies b4 is having_a_unity(b2)) & b5 | b3 = b6 | b3
   holds b4 $$(b3,b5) = b4 $$(b3,b6);

:: SETWOP_2:th 10
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4 being Element of b2
for b5 being Function-like quasi_total Relation of [:b2,b2:],b2
for b6 being Function-like quasi_total Relation of b1,b2
      st b5 is commutative(b2) & b5 is associative(b2) & b5 is having_a_unity(b2) & b4 = the_unity_wrt b5 & b6 .: b3 = {b4}
   holds b5 $$(b3,b6) = b4;

:: SETWOP_2:th 11
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4 being Element of b2
for b5, b6 being Function-like quasi_total Relation of [:b2,b2:],b2
for b7, b8 being Function-like quasi_total Relation of b1,b2
      st b5 is commutative(b2) &
         b5 is associative(b2) &
         b5 is having_a_unity(b2) &
         b4 = the_unity_wrt b5 &
         b6 .(b4,b4) = b4 &
         (for b9, b10, b11, b12 being Element of b2 holds
         b5 .(b6 .(b9,b10),b6 .(b11,b12)) = b6 .(b5 .(b9,b11),b5 .(b10,b12)))
   holds b6 .(b5 $$(b3,b7),b5 $$(b3,b8)) = b5 $$(b3,b6 .:(b7,b8));

:: SETWOP_2:th 12
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4 being Function-like quasi_total Relation of [:b2,b2:],b2
for b5, b6 being Function-like quasi_total Relation of b1,b2
      st b4 is commutative(b2) & b4 is associative(b2) & b4 is having_a_unity(b2)
   holds b4 .(b4 $$(b3,b5),b4 $$(b3,b6)) = b4 $$(b3,b4 .:(b5,b6));

:: SETWOP_2:th 13
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4, b5 being Function-like quasi_total Relation of [:b2,b2:],b2
for b6, b7 being Function-like quasi_total Relation of b1,b2
      st b4 is commutative(b2) & b4 is associative(b2) & b4 is having_a_unity(b2) & b4 is having_an_inverseOp(b2) & b5 = b4 *(id b2,the_inverseOp_wrt b4)
   holds b5 .(b4 $$(b3,b6),b4 $$(b3,b7)) = b4 $$(b3,b5 .:(b6,b7));

:: SETWOP_2:th 14
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4, b5 being Element of b2
for b6, b7 being Function-like quasi_total Relation of [:b2,b2:],b2
for b8 being Function-like quasi_total Relation of b1,b2
      st b6 is commutative(b2) & b6 is associative(b2) & b6 is having_a_unity(b2) & b4 = the_unity_wrt b6 & b7 is_distributive_wrt b6 & b7 .(b5,b4) = b4
   holds b7 .(b5,b6 $$(b3,b8)) = b6 $$(b3,b7 [;](b5,b8));

:: SETWOP_2:th 15
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4, b5 being Element of b2
for b6, b7 being Function-like quasi_total Relation of [:b2,b2:],b2
for b8 being Function-like quasi_total Relation of b1,b2
      st b6 is commutative(b2) & b6 is associative(b2) & b6 is having_a_unity(b2) & b4 = the_unity_wrt b6 & b7 is_distributive_wrt b6 & b7 .(b4,b5) = b4
   holds b7 .(b6 $$(b3,b8),b5) = b6 $$(b3,b7 [:](b8,b5));

:: SETWOP_2:th 16
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4 being Element of b2
for b5, b6 being Function-like quasi_total Relation of [:b2,b2:],b2
for b7 being Function-like quasi_total Relation of b1,b2
      st b5 is commutative(b2) & b5 is associative(b2) & b5 is having_a_unity(b2) & b5 is having_an_inverseOp(b2) & b6 is_distributive_wrt b5
   holds b6 .(b4,b5 $$(b3,b7)) = b5 $$(b3,b6 [;](b4,b7));

:: SETWOP_2:th 17
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4 being Element of b2
for b5, b6 being Function-like quasi_total Relation of [:b2,b2:],b2
for b7 being Function-like quasi_total Relation of b1,b2
      st b5 is commutative(b2) & b5 is associative(b2) & b5 is having_a_unity(b2) & b5 is having_an_inverseOp(b2) & b6 is_distributive_wrt b5
   holds b6 .(b5 $$(b3,b7),b4) = b5 $$(b3,b6 [:](b7,b4));

:: SETWOP_2:th 18
theorem
for b1, b2, b3 being non empty set
for b4 being Element of Fin b1
for b5 being Function-like quasi_total Relation of [:b3,b3:],b3
for b6 being Function-like quasi_total Relation of b1,b3
for b7 being Function-like quasi_total Relation of [:b2,b2:],b2
for b8 being Function-like quasi_total Relation of b3,b2
      st b5 is commutative(b3) &
         b5 is associative(b3) &
         b5 is having_a_unity(b3) &
         b7 is commutative(b2) &
         b7 is associative(b2) &
         b7 is having_a_unity(b2) &
         b8 . the_unity_wrt b5 = the_unity_wrt b7 &
         (for b9, b10 being Element of b3 holds
         b8 . (b5 .(b9,b10)) = b7 .(b8 . b9,b8 . b10))
   holds b8 . (b5 $$(b4,b6)) = b7 $$(b4,b8 * b6);

:: SETWOP_2:th 19
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4 being Function-like quasi_total Relation of [:b2,b2:],b2
for b5 being Function-like quasi_total Relation of b2,b2
for b6 being Function-like quasi_total Relation of b1,b2
      st b4 is commutative(b2) & b4 is associative(b2) & b4 is having_a_unity(b2) & b5 . the_unity_wrt b4 = the_unity_wrt b4 & b5 is_distributive_wrt b4
   holds b5 . (b4 $$(b3,b6)) = b4 $$(b3,b5 * b6);

:: SETWOP_2:th 20
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4 being Element of b2
for b5, b6 being Function-like quasi_total Relation of [:b2,b2:],b2
for b7 being Function-like quasi_total Relation of b1,b2
      st b5 is commutative(b2) & b5 is associative(b2) & b5 is having_a_unity(b2) & b5 is having_an_inverseOp(b2) & b6 is_distributive_wrt b5
   holds (b6 [;](b4,id b2)) . (b5 $$(b3,b7)) = b5 $$(b3,(b6 [;](b4,id b2)) * b7);

:: SETWOP_2:th 21
theorem
for b1, b2 being non empty set
for b3 being Element of Fin b1
for b4 being Function-like quasi_total Relation of [:b2,b2:],b2
for b5 being Function-like quasi_total Relation of b1,b2
      st b4 is commutative(b2) & b4 is associative(b2) & b4 is having_a_unity(b2) & b4 is having_an_inverseOp(b2)
   holds (the_inverseOp_wrt b4) . (b4 $$(b3,b5)) = b4 $$(b3,(the_inverseOp_wrt b4) * b5);

:: SETWOP_2:funcnot 1 => SETWOP_2:func 1
definition
  let a1 be non empty set;
  let a2 be FinSequence of a1;
  let a3 be Element of a1;
  func [#](A2,A3) -> Function-like quasi_total Relation of NAT,a1 equals
    (NAT --> a3) +* a2;
end;

:: SETWOP_2:def 1
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of b1 holds
   [#](b2,b3) = (NAT --> b3) +* b2;

:: SETWOP_2:th 22
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being natural set
for b4 being FinSequence of b1 holds
   (b3 in dom b4 implies ([#](b4,b2)) . b3 = b4 . b3) &
    (b3 in dom b4 or ([#](b4,b2)) . b3 = b2);

:: SETWOP_2:th 23
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being FinSequence of b1 holds
   ([#](b3,b2)) | dom b3 = b3;

:: SETWOP_2:th 24
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being FinSequence of b1 holds
([#](b3 ^ b4,b2)) | dom b3 = b3;

:: SETWOP_2:th 25
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being FinSequence of b1 holds
   proj2 [#](b3,b2) = (proj2 b3) \/ {b2};

:: SETWOP_2:th 26
theorem
for b1, b2 being non empty set
for b3 being Element of b2
for b4 being Function-like quasi_total Relation of b2,b1
for b5 being FinSequence of b2 holds
   b4 * [#](b5,b3) = [#](b4 * b5,b4 . b3);

:: SETWOP_2:funcnot 2 => FINSEQ_1:func 1
notation
  let a1 be natural set;
  synonym finSeg a1 for Seg a1;
end;

:: SETWOP_2:funcnot 3 => SETWOP_2:func 2
definition
  let a1 be natural set;
  redefine func finSeg a1 -> Element of Fin NAT;
end;

:: SETWOP_2:funcnot 4 => FINSOP_1:func 2
notation
  let a1 be non empty set;
  let a2 be FinSequence of a1;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
  synonym a3 $$ a2 for a3 "**" a2;
end;

:: SETWOP_2:funcnot 5 => FINSOP_1:func 2
definition
  let a1 be non empty set;
  let a2 be FinSequence of a1;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
  assume (a3 is having_a_unity(a1) or 1 <= len a2) & a3 is associative(a1) & a3 is commutative(a1);
  func A3 $$ A2 -> Element of a1 equals
    a3 $$(findom a2,[#](a2,the_unity_wrt a3));
end;

:: SETWOP_2:def 2
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1
      st (b3 is having_a_unity(b1) or 1 <= len b2) & b3 is associative(b1) & b3 is commutative(b1)
   holds b3 "**" b2 = b3 $$(findom b2,[#](b2,the_unity_wrt b3));

:: SETWOP_2:th 35
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
for b3 being natural set
      st b2 is having_a_unity(b1)
   holds b2 "**" (b3 |-> the_unity_wrt b2) = the_unity_wrt b2;

:: SETWOP_2:th 37
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1
for b4, b5 being natural set
      st b3 is associative(b1) &
         (1 <= b4 & 1 <= b5 or b3 is having_a_unity(b1))
   holds b3 "**" ((b4 + b5) |-> b2) = b3 .(b3 "**" (b4 |-> b2),b3 "**" (b5 |-> b2));

:: SETWOP_2:th 38
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1
for b4, b5 being natural set
      st b3 is commutative(b1) &
         b3 is associative(b1) &
         (1 <= b4 & 1 <= b5 or b3 is having_a_unity(b1))
   holds b3 "**" ((b4 * b5) |-> b2) = b3 "**" (b5 |-> (b3 "**" (b4 |-> b2)));

:: SETWOP_2:th 39
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of [:b2,b2:],b2
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
for b5 being Function-like quasi_total Relation of b2,b1
for b6 being FinSequence of b2
      st b3 is having_a_unity(b2) &
         b4 is having_a_unity(b1) &
         b5 . the_unity_wrt b3 = the_unity_wrt b4 &
         (for b7, b8 being Element of b2 holds
         b5 . (b3 .(b7,b8)) = b4 .(b5 . b7,b5 . b8))
   holds b5 . (b3 "**" b6) = b4 "**" (b5 * b6);

:: SETWOP_2:th 40
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
for b3 being Function-like quasi_total Relation of b1,b1
for b4 being FinSequence of b1
      st b2 is having_a_unity(b1) & b3 . the_unity_wrt b2 = the_unity_wrt b2 & b3 is_distributive_wrt b2
   holds b3 . (b2 "**" b4) = b2 "**" (b3 * b4);

:: SETWOP_2:th 41
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like quasi_total Relation of [:b1,b1:],b1
for b5 being FinSequence of b1
      st b3 is associative(b1) & b3 is having_a_unity(b1) & b3 is having_an_inverseOp(b1) & b4 is_distributive_wrt b3
   holds (b4 [;](b2,id b1)) . (b3 "**" b5) = b3 "**" ((b4 [;](b2,id b1)) * b5);

:: SETWOP_2:th 42
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
for b3 being FinSequence of b1
      st b2 is commutative(b1) & b2 is associative(b1) & b2 is having_a_unity(b1) & b2 is having_an_inverseOp(b1)
   holds (the_inverseOp_wrt b2) . (b2 "**" b3) = b2 "**" ((the_inverseOp_wrt b2) * b3);

:: SETWOP_2:th 43
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like quasi_total Relation of [:b1,b1:],b1
for b5, b6 being FinSequence of b1
      st b3 is commutative(b1) &
         b3 is associative(b1) &
         b3 is having_a_unity(b1) &
         b2 = the_unity_wrt b3 &
         b4 .(b2,b2) = b2 &
         (for b7, b8, b9, b10 being Element of b1 holds
         b3 .(b4 .(b7,b8),b4 .(b9,b10)) = b4 .(b3 .(b7,b9),b3 .(b8,b10))) &
         len b5 = len b6
   holds b4 .(b3 "**" b5,b3 "**" b6) = b3 "**" (b4 .:(b5,b6));

:: SETWOP_2:th 44
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like quasi_total Relation of [:b1,b1:],b1
for b5 being natural set
for b6, b7 being Element of b5 -tuples_on b1
      st b3 is commutative(b1) &
         b3 is associative(b1) &
         b3 is having_a_unity(b1) &
         b2 = the_unity_wrt b3 &
         b4 .(b2,b2) = b2 &
         (for b8, b9, b10, b11 being Element of b1 holds
         b3 .(b4 .(b8,b9),b4 .(b10,b11)) = b4 .(b3 .(b8,b10),b3 .(b9,b11)))
   holds b4 .(b3 "**" b6,b3 "**" b7) = b3 "**" (b4 .:(b6,b7));

:: SETWOP_2:th 45
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
for b3, b4 being FinSequence of b1
      st b2 is commutative(b1) & b2 is associative(b1) & b2 is having_a_unity(b1) & len b3 = len b4
   holds b2 .(b2 "**" b3,b2 "**" b4) = b2 "**" (b2 .:(b3,b4));

:: SETWOP_2:th 46
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
for b3 being natural set
for b4, b5 being Element of b3 -tuples_on b1
      st b2 is commutative(b1) & b2 is associative(b1) & b2 is having_a_unity(b1)
   holds b2 .(b2 "**" b4,b2 "**" b5) = b2 "**" (b2 .:(b4,b5));

:: SETWOP_2:th 47
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
for b5 being natural set
      st b4 is commutative(b1) & b4 is associative(b1) & b4 is having_a_unity(b1)
   holds b4 "**" (b5 |-> (b4 .(b2,b3))) = b4 .(b4 "**" (b5 |-> b2),b4 "**" (b5 |-> b3));

:: SETWOP_2:th 48
theorem
for b1 being non empty set
for b2, b3 being Function-like quasi_total Relation of [:b1,b1:],b1
for b4 being natural set
for b5, b6 being Element of b4 -tuples_on b1
      st b2 is commutative(b1) & b2 is associative(b1) & b2 is having_a_unity(b1) & b2 is having_an_inverseOp(b1) & b3 = b2 *(id b1,the_inverseOp_wrt b2)
   holds b3 .(b2 "**" b5,b2 "**" b6) = b2 "**" (b3 .:(b5,b6));

:: SETWOP_2:th 49
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4, b5 being Function-like quasi_total Relation of [:b1,b1:],b1
for b6 being FinSequence of b1
      st b4 is commutative(b1) & b4 is associative(b1) & b4 is having_a_unity(b1) & b2 = the_unity_wrt b4 & b5 is_distributive_wrt b4 & b5 .(b3,b2) = b2
   holds b5 .(b3,b4 "**" b6) = b4 "**" (b5 [;](b3,b6));

:: SETWOP_2:th 50
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4, b5 being Function-like quasi_total Relation of [:b1,b1:],b1
for b6 being FinSequence of b1
      st b4 is commutative(b1) & b4 is associative(b1) & b4 is having_a_unity(b1) & b2 = the_unity_wrt b4 & b5 is_distributive_wrt b4 & b5 .(b2,b3) = b2
   holds b5 .(b4 "**" b6,b3) = b4 "**" (b5 [:](b6,b3));

:: SETWOP_2:th 51
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like quasi_total Relation of [:b1,b1:],b1
for b5 being FinSequence of b1
      st b3 is commutative(b1) & b3 is associative(b1) & b3 is having_a_unity(b1) & b3 is having_an_inverseOp(b1) & b4 is_distributive_wrt b3
   holds b4 .(b2,b3 "**" b5) = b3 "**" (b4 [;](b2,b5));

:: SETWOP_2:th 52
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like quasi_total Relation of [:b1,b1:],b1
for b5 being FinSequence of b1
      st b3 is commutative(b1) & b3 is associative(b1) & b3 is having_a_unity(b1) & b3 is having_an_inverseOp(b1) & b4 is_distributive_wrt b3
   holds b4 .(b3 "**" b5,b2) = b3 "**" (b4 [:](b5,b2));