Article PRVECT_2, MML version 4.99.1005

:: PRVECT_2:th 1
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being real set
      st for b4 being Element of NAT holds
           b2 . b4 = abs ((b1 . b4) - b3)
   holds    b1 is convergent & lim b1 = b3
   iff
      b2 is convergent & lim b2 = 0;

:: PRVECT_2:th 2
theorem
for b1, b2 being FinSequence of REAL
      st len b1 = len b2 &
         (for b3 being Element of NAT
               st b3 in Seg len b1
            holds 0 <= b1 . b3 & b1 . b3 <= b2 . b3)
   holds |.b1.| <= |.b2.|;

:: PRVECT_2:th 3
theorem
for b1 being FinSequence of REAL
      st for b2 being Element of NAT
              st b2 in dom b1
           holds b1 . b2 = 0
   holds Sum b1 = 0;

:: PRVECT_2:modenot 1 => PRVECT_2:mode 1
definition
  let a1 be Relation-like Function-like set;
  let a2 be set;
  mode MultOps of A2,A1 -> Relation-like Function-like set means
    proj1 it = proj1 a1 &
     (for b1 being set
           st b1 in proj1 a1
        holds it . b1 is Function-like quasi_total Relation of [:a2,a1 . b1:],a1 . b1);
end;

:: PRVECT_2:dfs 1
definiens
  let a1 be Relation-like Function-like set;
  let a2 be set;
  let a3 be Relation-like Function-like set;
To prove
     a3 is MultOps of a2,a1
it is sufficient to prove
  thus proj1 a3 = proj1 a1 &
     (for b1 being set
           st b1 in proj1 a1
        holds a3 . b1 is Function-like quasi_total Relation of [:a2,a1 . b1:],a1 . b1);

:: PRVECT_2:def 1
theorem
for b1 being Relation-like Function-like set
for b2 being set
for b3 being Relation-like Function-like set holds
      b3 is MultOps of b2,b1
   iff
      proj1 b3 = proj1 b1 &
       (for b4 being set
             st b4 in proj1 b1
          holds b3 . b4 is Function-like quasi_total Relation of [:b2,b1 . b4:],b1 . b4);

:: PRVECT_2:condreg 1
registration
  let a1 be Relation-like non-empty Function-like non empty FinSequence-like set;
  let a2 be set;
  cluster -> FinSequence-like (MultOps of a2,a1);
end;

:: PRVECT_2:th 4
theorem
for b1 being set
for b2 being Relation-like non-empty Function-like non empty FinSequence-like set
for b3 being Relation-like Function-like FinSequence-like set holds
      b3 is MultOps of b1,b2
   iff
      len b3 = len b2 &
       (for b4 being set
             st b4 in dom b2
          holds b3 . b4 is Function-like quasi_total Relation of [:b1,b2 . b4:],b2 . b4);

:: PRVECT_2:funcnot 1 => PRVECT_2:func 1
definition
  let a1 be Relation-like non-empty Function-like non empty FinSequence-like set;
  let a2 be set;
  let a3 be MultOps of a2,a1;
  let a4 be Element of dom a1;
  redefine func a3 . a4 -> Function-like quasi_total Relation of [:a2,a1 . a4:],a1 . a4;
end;

:: PRVECT_2:th 5
theorem
for b1 being non empty set
for b2 being Relation-like non-empty Function-like non empty FinSequence-like set
for b3, b4 being Function-like quasi_total Relation of [:b1,product b2:],product b2
      st for b5 being Element of b1
        for b6 being Element of product b2
        for b7 being Element of dom b2 holds
           (b3 .(b5,b6)) . b7 = (b4 .(b5,b6)) . b7
   holds b3 = b4;

:: PRVECT_2:funcnot 2 => PRVECT_2:func 2
definition
  let a1 be Relation-like non-empty Function-like non empty FinSequence-like set;
  let a2 be non empty set;
  let a3 be MultOps of a2,a1;
  func [:A3:] -> Function-like quasi_total Relation of [:a2,product a1:],product a1 means
    for b1 being Element of a2
    for b2 being Element of product a1
    for b3 being Element of dom a1 holds
       (it .(b1,b2)) . b3 = (a3 . b3) .(b1,b2 . b3);
end;

:: PRVECT_2:def 2
theorem
for b1 being Relation-like non-empty Function-like non empty FinSequence-like set
for b2 being non empty set
for b3 being MultOps of b2,b1
for b4 being Function-like quasi_total Relation of [:b2,product b1:],product b1 holds
      b4 = [:b3:]
   iff
      for b5 being Element of b2
      for b6 being Element of product b1
      for b7 being Element of dom b1 holds
         (b4 .(b5,b6)) . b7 = (b3 . b7) .(b5,b6 . b7);

:: PRVECT_2:attrnot 1 => PRVECT_2:attr 1
definition
  let a1 be Relation-like set;
  attr a1 is RealLinearSpace-yielding means
    for b1 being set
          st b1 in proj2 a1
       holds b1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
end;

:: PRVECT_2:dfs 3
definiens
  let a1 be Relation-like set;
To prove
     a1 is RealLinearSpace-yielding
it is sufficient to prove
  thus for b1 being set
          st b1 in proj2 a1
       holds b1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;

:: PRVECT_2:def 3
theorem
for b1 being Relation-like set holds
      b1 is RealLinearSpace-yielding
   iff
      for b2 being set
            st b2 in proj2 b1
         holds b2 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;

:: PRVECT_2:exreg 1
registration
  cluster Relation-like Function-like non empty finite FinSequence-like RealLinearSpace-yielding set;
end;

:: PRVECT_2:modenot 2
definition
  mode RealLinearSpace-Sequence is Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set;
end;

:: PRVECT_2:funcnot 3 => PRVECT_2:func 3
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set;
  let a2 be Element of dom a1;
  redefine func a1 . a2 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
end;

:: PRVECT_2:funcnot 4 => PRVECT_2:func 4
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set;
  func carr A1 -> Relation-like non-empty Function-like non empty FinSequence-like set means
    len it = len a1 &
     (for b1 being Element of dom a1 holds
        it . b1 = the carrier of a1 . b1);
end;

:: PRVECT_2:def 4
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set
for b2 being Relation-like non-empty Function-like non empty FinSequence-like set holds
      b2 = carr b1
   iff
      len b2 = len b1 &
       (for b3 being Element of dom b1 holds
          b2 . b3 = the carrier of b1 . b3);

:: PRVECT_2:funcnot 5 => PRVECT_2:func 5
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set;
  let a2 be Element of dom carr a1;
  redefine func a1 . a2 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
end;

:: PRVECT_2:funcnot 6 => PRVECT_2:func 6
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set;
  func addop A1 -> BinOps of carr a1 means
    len it = len carr a1 &
     (for b1 being Element of dom carr a1 holds
        it . b1 = the addF of a1 . b1);
end;

:: PRVECT_2:def 5
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set
for b2 being BinOps of carr b1 holds
      b2 = addop b1
   iff
      len b2 = len carr b1 &
       (for b3 being Element of dom carr b1 holds
          b2 . b3 = the addF of b1 . b3);

:: PRVECT_2:funcnot 7 => PRVECT_2:func 7
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set;
  func complop A1 -> UnOps of carr a1 means
    len it = len carr a1 &
     (for b1 being Element of dom carr a1 holds
        it . b1 = comp (a1 . b1));
end;

:: PRVECT_2:def 6
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set
for b2 being UnOps of carr b1 holds
      b2 = complop b1
   iff
      len b2 = len carr b1 &
       (for b3 being Element of dom carr b1 holds
          b2 . b3 = comp (b1 . b3));

:: PRVECT_2:funcnot 8 => PRVECT_2:func 8
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set;
  func zeros A1 -> Element of product carr a1 means
    for b1 being Element of dom carr a1 holds
       it . b1 = the ZeroF of a1 . b1;
end;

:: PRVECT_2:def 7
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set
for b2 being Element of product carr b1 holds
      b2 = zeros b1
   iff
      for b3 being Element of dom carr b1 holds
         b2 . b3 = the ZeroF of b1 . b3;

:: PRVECT_2:funcnot 9 => PRVECT_2:func 9
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set;
  func multop A1 -> MultOps of REAL,carr a1 means
    len it = len carr a1 &
     (for b1 being Element of dom carr a1 holds
        it . b1 = the Mult of a1 . b1);
end;

:: PRVECT_2:def 8
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set
for b2 being MultOps of REAL,carr b1 holds
      b2 = multop b1
   iff
      len b2 = len carr b1 &
       (for b3 being Element of dom carr b1 holds
          b2 . b3 = the Mult of b1 . b3);

:: PRVECT_2:funcnot 10 => PRVECT_2:func 10
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set;
  func product A1 -> non empty strict RLSStruct equals
    RLSStruct(#product carr a1,zeros a1,[:addop a1:],[:multop a1:]#);
end;

:: PRVECT_2:def 9
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set holds
   product b1 = RLSStruct(#product carr b1,zeros b1,[:addop b1:],[:multop b1:]#);

:: PRVECT_2:funcreg 1
registration
  let a1 be Relation-like Function-like non empty FinSequence-like RealLinearSpace-yielding set;
  cluster product a1 -> non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like;
end;

:: PRVECT_2:attrnot 2 => PRVECT_2:attr 2
definition
  let a1 be Relation-like set;
  attr a1 is RealNormSpace-yielding means
    for b1 being set
          st b1 in proj2 a1
       holds b1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
end;

:: PRVECT_2:dfs 10
definiens
  let a1 be Relation-like set;
To prove
     a1 is RealNormSpace-yielding
it is sufficient to prove
  thus for b1 being set
          st b1 in proj2 a1
       holds b1 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;

:: PRVECT_2:def 10
theorem
for b1 being Relation-like set holds
      b1 is RealNormSpace-yielding
   iff
      for b2 being set
            st b2 in proj2 b1
         holds b2 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;

:: PRVECT_2:exreg 2
registration
  cluster Relation-like Function-like non empty finite FinSequence-like RealNormSpace-yielding set;
end;

:: PRVECT_2:modenot 3
definition
  mode RealNormSpace-Sequence is Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set;
end;

:: PRVECT_2:funcnot 11 => PRVECT_2:func 11
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set;
  let a2 be Element of dom a1;
  redefine func a1 . a2 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
end;

:: PRVECT_2:condreg 2
registration
  cluster Relation-like Function-like FinSequence-like RealNormSpace-yielding -> RealLinearSpace-yielding (set);
end;

:: PRVECT_2:funcnot 12 => PRVECT_2:func 12
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set;
  let a2 be Element of product carr a1;
  func normsequence(A1,A2) -> Element of REAL len a1 means
    len it = len a1 &
     (for b1 being Element of dom a1 holds
        it . b1 = (the norm of a1 . b1) . (a2 . b1));
end;

:: PRVECT_2:def 11
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set
for b2 being Element of product carr b1
for b3 being Element of REAL len b1 holds
      b3 = normsequence(b1,b2)
   iff
      len b3 = len b1 &
       (for b4 being Element of dom b1 holds
          b3 . b4 = (the norm of b1 . b4) . (b2 . b4));

:: PRVECT_2:funcnot 13 => PRVECT_2:func 13
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set;
  func productnorm A1 -> Function-like quasi_total Relation of product carr a1,REAL means
    for b1 being Element of product carr a1 holds
       it . b1 = |.normsequence(a1,b1).|;
end;

:: PRVECT_2:def 12
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set
for b2 being Function-like quasi_total Relation of product carr b1,REAL holds
      b2 = productnorm b1
   iff
      for b3 being Element of product carr b1 holds
         b2 . b3 = |.normsequence(b1,b3).|;

:: PRVECT_2:funcnot 14 => PRVECT_2:func 14
definition
  let a1 be Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set;
  func product A1 -> non empty strict NORMSTR means
    RLSStruct(#the carrier of it,the ZeroF of it,the addF of it,the Mult of it#) = product a1 &
     the norm of it = productnorm a1;
end;

:: PRVECT_2:def 13
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set
for b2 being non empty strict NORMSTR holds
      b2 = product b1
   iff
      RLSStruct(#the carrier of b2,the ZeroF of b2,the addF of b2,the Mult of b2#) = product b1 &
       the norm of b2 = productnorm b1;

:: PRVECT_2:th 6
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set holds
   product b1 = NORMSTR(#product carr b1,zeros b1,[:addop b1:],[:multop b1:],productnorm b1#);

:: PRVECT_2:th 7
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set
for b2 being Element of the carrier of product b1
for b3 being Element of product carr b1
      st b2 = b3
   holds ||.b2.|| = |.normsequence(b1,b3).|;

:: PRVECT_2:th 8
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set
for b2, b3, b4 being Element of product carr b1
for b5 being Element of NAT
      st b5 in proj1 b2 & b4 = [:addop b1:] .(b2,b3)
   holds (normsequence(b1,b4)) . b5 <= ((normsequence(b1,b2)) + normsequence(b1,b3)) . b5;

:: PRVECT_2:th 9
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set
for b2 being Element of product carr b1
for b3 being Element of NAT
      st b3 in proj1 b2
   holds 0 <= (normsequence(b1,b2)) . b3;

:: PRVECT_2:funcreg 2
registration
  let a1 be Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set;
  cluster product a1 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealNormSpace-like;
end;

:: PRVECT_2:th 10
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set
for b2 being Element of dom b1
for b3 being Element of the carrier of product b1
for b4 being Element of product carr b1
for b5 being Element of the carrier of b1 . b2
      st b4 = b3 & b5 = b4 . b2
   holds ||.b5.|| <= ||.b3.||;

:: PRVECT_2:th 11
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set
for b2 being Element of dom b1
for b3, b4 being Element of the carrier of product b1
for b5, b6 being Element of the carrier of b1 . b2
for b7, b8 being Element of product carr b1
      st b5 = b7 . b2 & b7 = b3 & b6 = b8 . b2 & b8 = b4
   holds ||.b6 - b5.|| <= ||.b4 - b3.||;

:: PRVECT_2:th 12
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set
for b2 being Function-like quasi_total Relation of NAT,the carrier of product b1
for b3 being Element of the carrier of product b1
for b4 being Element of product carr b1
   st b3 = b4 & b2 is convergent(product b1) & lim b2 = b3
for b5 being Element of dom b1 holds
   ex b6 being Function-like quasi_total Relation of NAT,the carrier of b1 . b5 st
      b6 is convergent(b1 . b5) &
       b4 . b5 = lim b6 &
       (for b7 being Element of NAT holds
          ex b8 being Element of product carr b1 st
             b8 = b2 . b7 & b6 . b7 = b8 . b5);

:: PRVECT_2:th 13
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set
for b2 being Function-like quasi_total Relation of NAT,the carrier of product b1
for b3 being Element of the carrier of product b1
for b4 being Element of product carr b1
      st b3 = b4 &
         (for b5 being Element of dom b1 holds
            ex b6 being Function-like quasi_total Relation of NAT,the carrier of b1 . b5 st
               b6 is convergent(b1 . b5) &
                b4 . b5 = lim b6 &
                (for b7 being Element of NAT holds
                   ex b8 being Element of product carr b1 st
                      b8 = b2 . b7 & b6 . b7 = b8 . b5))
   holds b2 is convergent(product b1) & lim b2 = b3;

:: PRVECT_2:th 14
theorem
for b1 being Relation-like Function-like non empty FinSequence-like RealNormSpace-yielding set
      st for b2 being Element of dom b1 holds
           b1 . b2 is complete
   holds product b1 is complete;