Article SEQ_1, MML version 4.99.1005

:: SEQ_1:modenot 1
definition
  mode Real_Sequence is Function-like quasi_total Relation of NAT,REAL;
end;

:: SEQ_1:th 3
theorem
for b1 being Relation-like Function-like set holds
      b1 is Function-like quasi_total Relation of NAT,REAL
   iff
      proj1 b1 = NAT &
       (for b2 being set
             st b2 in NAT
          holds b1 . b2 is real);

:: SEQ_1:th 4
theorem
for b1 being Relation-like Function-like set holds
      b1 is Function-like quasi_total Relation of NAT,REAL
   iff
      proj1 b1 = NAT &
       (for b2 being Element of NAT holds
          b1 . b2 is real);

:: SEQ_1:attrnot 1 => VALUED_0:attr 3
notation
  let a1 be Relation-like set;
  synonym real-yielding for real-valued;
end;

:: SEQ_1:funcnot 1 => SEQ_1:func 1
definition
  let a1 be Relation-like Function-like real-valued set;
  let a2 be set;
  redefine func a1 . a2 -> Element of REAL;
end;

:: SEQ_1:attrnot 2 => RELAT_1:attr 2
notation
  let a1 be Function-like Relation of NAT,REAL;
  synonym being_not_0 for non-empty;
end;

:: SEQ_1:prednot 1 => RELAT_1:attr 2
notation
  let a1 be Function-like Relation of NAT,REAL;
  synonym a1 is_not_0 for non-empty;
end;

:: SEQ_1:attrnot 3 => RELAT_1:attr 2
definition
  let a1 be Relation-like set;
  attr a1 is being_not_0 means
    rng a1 c= REAL \ {0};
end;

:: SEQ_1:dfs 1
definiens
  let a1 be Function-like Relation of NAT,REAL;
To prove
     a1 is non-empty
it is sufficient to prove
  thus rng a1 c= REAL \ {0};

:: SEQ_1:def 2
theorem
for b1 being Function-like Relation of NAT,REAL holds
      b1 is non-empty
   iff
      rng b1 c= REAL \ {0};

:: SEQ_1:th 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is non-empty
   iff
      for b2 being set
            st b2 in NAT
         holds b1 . b2 <> 0;

:: SEQ_1:th 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is non-empty
   iff
      for b2 being Element of NAT holds
         b1 . b2 <> 0;

:: SEQ_1:th 8
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being set
              st b3 in NAT
           holds b1 . b3 = b2 . b3
   holds b1 = b2;

:: SEQ_1:th 10
theorem
for b1 being real set holds
   ex b2 being Function-like quasi_total Relation of NAT,REAL st
      rng b2 = {b1};

:: SEQ_1:sch 1
scheme SEQ_1:sch 1
{F1 -> real set}:
ex b1 being Function-like quasi_total Relation of NAT,REAL st
   for b2 being Element of NAT holds
      b1 . b2 = F1(b2)


:: SEQ_1:sch 2
scheme SEQ_1:sch 2
{F1 -> non empty set,
  F2 -> non empty set}:
ex b1 being Function-like Relation of F1(),F2() st
   (for b2 being Element of F1() holds
          b2 in dom b1
       iff
          ex b3 being Element of F2() st
             P1[b2, b3]) &
    (for b2 being Element of F1()
          st b2 in dom b1
       holds P1[b2, b1 . b2])


:: SEQ_1:sch 3
scheme SEQ_1:sch 3
{F1 -> non empty set,
  F2 -> non empty set,
  F3 -> Element of F2()}:
ex b1 being Function-like Relation of F1(),F2() st
   (for b2 being Element of F1() holds
          b2 in dom b1
       iff
          P1[b2]) &
    (for b2 being Element of F1()
          st b2 in dom b1
       holds b1 . b2 = F3(b2))


:: SEQ_1:sch 4
scheme SEQ_1:sch 4
{F1 -> set,
  F2 -> set,
  F3 -> set,
  F4 -> set}:
for b1, b2 being Function-like Relation of F1(),F2()
      st dom b1 = F3() &
         (for b3 being Element of F1()
               st b3 in dom b1
            holds b1 . b3 = F4(b3)) &
         dom b2 = F3() &
         (for b3 being Element of F1()
               st b3 in dom b2
            holds b2 . b3 = F4(b3))
   holds b1 = b2


:: SEQ_1:th 11
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
   b1 = b2 + b3
iff
   for b4 being Element of NAT holds
      b1 . b4 = (b2 . b4) + (b3 . b4);

:: SEQ_1:th 12
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
   b1 = b2 (#) b3
iff
   for b4 being Element of NAT holds
      b1 . b4 = (b2 . b4) * (b3 . b4);

:: SEQ_1:th 13
theorem
for b1 being real set
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
   b2 = b1 (#) b3
iff
   for b4 being Element of NAT holds
      b2 . b4 = b1 * (b3 . b4);

:: SEQ_1:th 14
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
   b1 = - b2
iff
   for b3 being Element of NAT holds
      b1 . b3 = - (b2 . b3);

:: SEQ_1:th 15
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
b1 - b2 = b1 + - b2;

:: SEQ_1:th 16
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
   b1 = abs b2
iff
   for b3 being Element of NAT holds
      b1 . b3 = abs (b2 . b3);

:: SEQ_1:th 20
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b1 + b2) + b3 = b1 + (b2 + b3);

:: SEQ_1:th 22
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b1 (#) b2) (#) b3 = b1 (#) (b2 (#) b3);

:: SEQ_1:th 23
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b1 + b2) (#) b3 = (b1 (#) b3) + (b2 (#) b3);

:: SEQ_1:th 24
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
b1 (#) (b2 + b3) = (b1 (#) b2) + (b1 (#) b3);

:: SEQ_1:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   - b1 = (- 1) (#) b1;

:: SEQ_1:th 26
theorem
for b1 being real set
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
b1 (#) (b2 (#) b3) = (b1 (#) b2) (#) b3;

:: SEQ_1:th 27
theorem
for b1 being real set
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
b1 (#) (b2 (#) b3) = b2 (#) (b1 (#) b3);

:: SEQ_1:th 28
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b1 - b2) (#) b3 = (b1 (#) b3) - (b2 (#) b3);

:: SEQ_1:th 29
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b1 (#) b2) - (b1 (#) b3) = b1 (#) (b2 - b3);

:: SEQ_1:th 30
theorem
for b1 being real set
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
b1 (#) (b2 + b3) = (b1 (#) b2) + (b1 (#) b3);

:: SEQ_1:th 31
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL holds
   (b1 * b2) (#) b3 = b1 (#) (b2 (#) b3);

:: SEQ_1:th 32
theorem
for b1 being real set
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
b1 (#) (b2 - b3) = (b1 (#) b2) - (b1 (#) b3);

:: SEQ_1:th 33
theorem
for b1 being real set
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
b1 (#) (b2 /" b3) = (b1 (#) b2) /" b3;

:: SEQ_1:th 34
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
b1 - (b2 + b3) = (b1 - b2) - b3;

:: SEQ_1:th 35
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   1 (#) b1 = b1;

:: SEQ_1:th 36
theorem
for b1 being Relation-like Function-like real-valued set holds
   - - b1 = b1;

:: SEQ_1:th 37
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
b1 - - b2 = b1 + b2;

:: SEQ_1:th 38
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
b1 - (b2 - b3) = (b1 - b2) + b3;

:: SEQ_1:th 39
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
b1 + (b2 - b3) = (b1 + b2) - b3;

:: SEQ_1:th 40
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(- b1) (#) b2 = - (b1 (#) b2) & b1 (#) - b2 = - (b1 (#) b2);

:: SEQ_1:th 41
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty
   holds b1 " is non-empty;

:: SEQ_1:th 42
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   b1 " " = b1;

:: SEQ_1:th 43
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
   b1 is non-empty & b2 is non-empty
iff
   b1 (#) b2 is non-empty;

:: SEQ_1:th 44
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
b1 " (#) (b2 ") = (b1 (#) b2) ";

:: SEQ_1:th 45
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty
   holds (b2 /" b1) (#) b1 = b2;

:: SEQ_1:th 46
theorem
for b1, b2, b3, b4 being Function-like quasi_total Relation of NAT,REAL holds
(b1 /" b2) (#) (b3 /" b4) = (b1 (#) b3) /" (b2 (#) b4);

:: SEQ_1:th 47
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty & b2 is non-empty
   holds b1 /" b2 is non-empty;

:: SEQ_1:th 48
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 /" b2) " = b2 /" b1;

:: SEQ_1:th 49
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
b1 (#) (b2 /" b3) = (b1 (#) b2) /" b3;

:: SEQ_1:th 50
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
b1 /" (b2 /" b3) = (b1 (#) b3) /" b2;

:: SEQ_1:th 51
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty
   holds b2 /" b3 = (b2 (#) b1) /" (b3 (#) b1);

:: SEQ_1:th 52
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 <> 0 & b2 is non-empty
   holds b1 (#) b2 is non-empty;

:: SEQ_1:th 53
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty
   holds - b1 is non-empty;

:: SEQ_1:th 54
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (b1 (#) b2) " = b1 " (#) (b2 ");

:: SEQ_1:th 55
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   (- b1) " = (- 1) (#) (b1 ");

:: SEQ_1:th 56
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
- (b1 /" b2) = (- b1) /" b2 & b1 /" - b2 = - (b1 /" b2);

:: SEQ_1:th 57
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b1 /" b2) + (b3 /" b2) = (b1 + b3) /" b2 &
 (b1 /" b2) - (b3 /" b2) = (b1 - b3) /" b2;

:: SEQ_1:th 58
theorem
for b1, b2, b3, b4 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty & b2 is non-empty
   holds (b3 /" b1) + (b4 /" b2) = ((b3 (#) b2) + (b4 (#) b1)) /" (b1 (#) b2) &
    (b3 /" b1) - (b4 /" b2) = ((b3 (#) b2) - (b4 (#) b1)) /" (b1 (#) b2);

:: SEQ_1:th 59
theorem
for b1, b2, b3, b4 being Function-like quasi_total Relation of NAT,REAL holds
(b1 /" b2) /" (b3 /" b4) = (b1 (#) b4) /" (b2 (#) b3);

:: SEQ_1:th 60
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
abs (b1 (#) b2) = (abs b1) (#) abs b2;

:: SEQ_1:th 61
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-empty
   holds abs b1 is non-empty;

:: SEQ_1:th 62
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   (abs b1) " = abs (b1 ");

:: SEQ_1:th 63
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
abs (b1 /" b2) = (abs b1) /" abs b2;

:: SEQ_1:th 64
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   abs (b1 (#) b2) = (abs b1) (#) abs b2;

:: SEQ_1:funcnot 2 => SEQ_1:func 2
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of NAT,a1;
  let a3 be natural set;
  redefine func a2 . a3 -> Element of a1;
end;