Article REAL_NS1, MML version 4.99.1005

:: REAL_NS1:funcnot 1 => REAL_NS1:func 1
definition
  let a1 be Element of NAT;
  func Euclid_add A1 -> Function-like quasi_total Relation of [:REAL a1,REAL a1:],REAL a1 means
    for b1, b2 being Element of REAL a1 holds
    it .(b1,b2) = b1 + b2;
end;

:: REAL_NS1:def 1
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of [:REAL b1,REAL b1:],REAL b1 holds
      b2 = Euclid_add b1
   iff
      for b3, b4 being Element of REAL b1 holds
      b2 .(b3,b4) = b3 + b4;

:: REAL_NS1:funcreg 1
registration
  let a1 be Element of NAT;
  cluster Euclid_add a1 -> Function-like quasi_total commutative associative;
end;

:: REAL_NS1:funcnot 2 => REAL_NS1:func 2
definition
  let a1 be Element of NAT;
  func Euclid_mult A1 -> Function-like quasi_total Relation of [:REAL,REAL a1:],REAL a1 means
    for b1 being Element of REAL
    for b2 being Element of REAL a1 holds
       it .(b1,b2) = b1 * b2;
end;

:: REAL_NS1:def 2
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of [:REAL,REAL b1:],REAL b1 holds
      b2 = Euclid_mult b1
   iff
      for b3 being Element of REAL
      for b4 being Element of REAL b1 holds
         b2 .(b3,b4) = b3 * b4;

:: REAL_NS1:funcnot 3 => REAL_NS1:func 3
definition
  let a1 be Element of NAT;
  func Euclid_norm A1 -> Function-like quasi_total Relation of REAL a1,REAL means
    for b1 being Element of REAL a1 holds
       it . b1 = |.b1.|;
end;

:: REAL_NS1:def 3
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of REAL b1,REAL holds
      b2 = Euclid_norm b1
   iff
      for b3 being Element of REAL b1 holds
         b2 . b3 = |.b3.|;

:: REAL_NS1:funcnot 4 => REAL_NS1:func 4
definition
  let a1 be Element of NAT;
  func REAL-NS A1 -> non empty strict NORMSTR means
    the carrier of it = REAL a1 & 0. it = 0* a1 & the addF of it = Euclid_add a1 & the Mult of it = Euclid_mult a1 & the norm of it = Euclid_norm a1;
end;

:: REAL_NS1:def 4
theorem
for b1 being Element of NAT
for b2 being non empty strict NORMSTR holds
      b2 = REAL-NS b1
   iff
      the carrier of b2 = REAL b1 & 0. b2 = 0* b1 & the addF of b2 = Euclid_add b1 & the Mult of b2 = Euclid_mult b1 & the norm of b2 = Euclid_norm b1;

:: REAL_NS1:funcreg 2
registration
  let a1 be non empty Element of NAT;
  cluster REAL-NS a1 -> non empty non trivial strict;
end;

:: REAL_NS1:th 1
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of REAL-NS b1
for b3 being Element of REAL b1
      st b2 = b3
   holds ||.b2.|| = |.b3.|;

:: REAL_NS1:th 2
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of REAL-NS b1
for b4, b5 being Element of REAL b1
      st b2 = b4 & b3 = b5
   holds b2 + b3 = b4 + b5;

:: REAL_NS1:th 3
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of REAL-NS b1
for b3 being Element of REAL b1
for b4 being Element of REAL
      st b2 = b3
   holds b4 * b2 = b4 * b3;

:: REAL_NS1:funcreg 3
registration
  let a1 be Element of NAT;
  cluster REAL-NS a1 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealNormSpace-like;
end;

:: REAL_NS1:th 4
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of REAL-NS b1
for b3 being Element of REAL b1
      st b2 = b3
   holds - b2 = - b3;

:: REAL_NS1:th 5
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of REAL-NS b1
for b4, b5 being Element of REAL b1
      st b2 = b4 & b3 = b5
   holds b2 - b3 = b4 - b5;

:: REAL_NS1:th 6
theorem
for b1 being Element of NAT
for b2 being FinSequence of REAL
      st dom b2 = Seg b1
   holds b2 is Element of REAL b1;

:: REAL_NS1:th 7
theorem
for b1 being Element of NAT
for b2 being Element of REAL b1
      st for b3 being Element of NAT
              st b3 in Seg b1
           holds 0 <= b2 . b3
   holds 0 <= Sum b2 &
    (for b3 being Element of NAT
          st b3 in Seg b1
       holds b2 . b3 <= Sum b2);

:: REAL_NS1:th 8
theorem
for b1 being Element of NAT
for b2 being Element of REAL b1
for b3 being Element of NAT
      st b3 in Seg b1
   holds abs (b2 . b3) <= |.b2.|;

:: REAL_NS1:th 9
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of REAL-NS b1
for b3 being Element of REAL b1
   st b2 = b3
for b4 being Element of NAT
      st b4 in Seg b1
   holds abs (b3 . b4) <= ||.b2.||;

:: REAL_NS1:th 10
theorem
for b1 being Element of NAT
for b2 being Element of REAL (b1 + 1) holds
   |.b2.| ^2 = |.b2 | b1.| ^2 + ((b2 . (b1 + 1)) ^2);

:: REAL_NS1:funcnot 5 => REAL_NS1:func 5
definition
  let a1 be Element of NAT;
  let a2 be Function-like quasi_total Relation of NAT,REAL a1;
  let a3 be Element of NAT;
  redefine func a2 . a3 -> Element of REAL a1;
end;

:: REAL_NS1:th 11
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of REAL-NS b1
for b3 being Element of REAL b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of REAL-NS b1
for b5 being Function-like quasi_total Relation of NAT,REAL b1
      st b3 = b2 & b5 = b4
   holds    b4 is convergent(REAL-NS b1) & lim b4 = b2
   iff
      for b6 being Element of NAT
            st b6 in Seg b1
         holds ex b7 being Function-like quasi_total Relation of NAT,REAL st
            for b8 being Element of NAT holds
               b7 . b8 = (b5 . b8) . b6 & b7 is convergent & b3 . b6 = lim b7;

:: REAL_NS1:th 12
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of REAL-NS b1
      st b2 is CCauchy(REAL-NS b1)
   holds b2 is convergent(REAL-NS b1);

:: REAL_NS1:funcreg 4
registration
  let a1 be Element of NAT;
  cluster REAL-NS a1 -> non empty strict complete;
end;

:: REAL_NS1:funcnot 6 => REAL_NS1:func 6
definition
  let a1 be Element of NAT;
  func Euclid_scalar A1 -> Function-like quasi_total Relation of [:REAL a1,REAL a1:],REAL means
    for b1, b2 being Element of REAL a1 holds
    it .(b1,b2) = Sum mlt(b1,b2);
end;

:: REAL_NS1:def 5
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of [:REAL b1,REAL b1:],REAL holds
      b2 = Euclid_scalar b1
   iff
      for b3, b4 being Element of REAL b1 holds
      b2 .(b3,b4) = Sum mlt(b3,b4);

:: REAL_NS1:funcnot 7 => REAL_NS1:func 7
definition
  let a1 be Element of NAT;
  func REAL-US A1 -> non empty strict UNITSTR means
    the carrier of it = REAL a1 & 0. it = 0* a1 & the addF of it = Euclid_add a1 & the Mult of it = Euclid_mult a1 & the scalar of it = Euclid_scalar a1;
end;

:: REAL_NS1:def 6
theorem
for b1 being Element of NAT
for b2 being non empty strict UNITSTR holds
      b2 = REAL-US b1
   iff
      the carrier of b2 = REAL b1 & 0. b2 = 0* b1 & the addF of b2 = Euclid_add b1 & the Mult of b2 = Euclid_mult b1 & the scalar of b2 = Euclid_scalar b1;

:: REAL_NS1:funcreg 5
registration
  let a1 be non empty Element of NAT;
  cluster REAL-US a1 -> non empty non trivial strict;
end;

:: REAL_NS1:funcreg 6
registration
  let a1 be Element of NAT;
  cluster REAL-US a1 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like;
end;

:: REAL_NS1:th 13
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3, b4 being Element of the carrier of REAL-NS b1
for b5, b6 being Element of the carrier of REAL-US b1
      st b3 = b5 & b4 = b6
   holds b3 + b4 = b5 + b6 & - b3 = - b5 & b2 * b3 = b2 * b5;

:: REAL_NS1:th 14
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of REAL-NS b1
for b3 being Element of the carrier of REAL-US b1
      st b2 = b3
   holds ||.b2.|| ^2 = b3 .|. b3;

:: REAL_NS1:th 15
theorem
for b1 being Element of NAT
for b2 being set holds
      b2 is Function-like quasi_total Relation of NAT,the carrier of REAL-NS b1
   iff
      b2 is Function-like quasi_total Relation of NAT,the carrier of REAL-US b1;

:: REAL_NS1:th 16
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of REAL-NS b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of REAL-US b1
      st b2 = b3
   holds (b2 is convergent(REAL-NS b1) implies b3 is convergent(REAL-US b1) & lim b2 = lim b3) &
    (b3 is convergent(REAL-US b1) implies b2 is convergent(REAL-NS b1) & lim b2 = lim b3);

:: REAL_NS1:th 17
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of REAL-NS b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of REAL-US b1
      st b2 = b3 & b2 is CCauchy(REAL-NS b1)
   holds b3 is Cauchy(REAL-US b1);

:: REAL_NS1:th 18
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,the carrier of REAL-NS b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of REAL-US b1
      st b2 = b3 & b3 is Cauchy(REAL-US b1)
   holds b2 is CCauchy(REAL-NS b1);

:: REAL_NS1:funcreg 7
registration
  let a1 be Element of NAT;
  cluster REAL-US a1 -> non empty strict Hilbert;
end;