Article JORDAN2B, MML version 4.99.1005

:: JORDAN2B:funcnot 1 => JORDAN2B:func 1
definition
  let a1, a2 be Element of NAT;
  let a3 be Element of the carrier of TOP-REAL a1;
  func Proj(A3,A2) -> Element of REAL means
    for b1 being FinSequence of REAL
          st b1 = a3
       holds it = b1 /. a2;
end;

:: JORDAN2B:def 1
theorem
for b1, b2 being Element of NAT
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of REAL holds
      b4 = Proj(b3,b2)
   iff
      for b5 being FinSequence of REAL
            st b5 = b3
         holds b4 = b5 /. b2;

:: JORDAN2B:th 1
theorem
for b1, b2 being Element of NAT holds
ex b3 being Function-like quasi_total Relation of the carrier of TOP-REAL b1,the carrier of R^1 st
   for b4 being Element of the carrier of TOP-REAL b1 holds
      b3 . b4 = Proj(b4,b2);

:: JORDAN2B:th 3
theorem
for b1, b2 being Element of NAT
      st b2 in Seg b1
   holds Proj(0.REAL b1,b2) = 0;

:: JORDAN2B:th 4
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of NAT
      st b4 in Seg b1
   holds Proj(b2 * b3,b4) = b2 * Proj(b3,b4);

:: JORDAN2B:th 5
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being Element of NAT
      st b3 in Seg b1
   holds Proj(- b2,b3) = - Proj(b2,b3);

:: JORDAN2B:th 6
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of NAT
      st b4 in Seg b1
   holds Proj(b2 + b3,b4) = (Proj(b2,b4)) + Proj(b3,b4);

:: JORDAN2B:th 7
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of NAT
      st b4 in Seg b1
   holds Proj(b2 - b3,b4) = (Proj(b2,b4)) - Proj(b3,b4);

:: JORDAN2B:th 9
theorem
for b1, b2 being Element of NAT
      st b1 <= b2
   holds (0* b2) | b1 = 0* b1;

:: JORDAN2B:th 10
theorem
for b1, b2 being Element of NAT holds
(0* b1) /^ b2 = 0* (b1 -' b2);

:: JORDAN2B:th 11
theorem
for b1 being Element of NAT holds
   Sum 0* b1 = 0;

:: JORDAN2B:th 12
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
for b3 being Element of NAT holds
   len (b1 +*(b3,b2)) = len b1;

:: JORDAN2B:th 13
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being FinSequence of b2
for b4 being Element of b2
      st b1 in dom b3
   holds b3 +*(b1,b4) = ((b3 | (b1 -' 1)) ^ <*b4*>) ^ (b3 /^ b1);

:: JORDAN2B:th 14
theorem
for b1, b2 being Element of NAT
for b3 being Element of REAL
      st b1 in Seg b2
   holds Sum ((0* b2) +*(b1,b3)) = b3;

:: JORDAN2B:th 15
theorem
for b1 being Element of NAT
for b2 being Element of REAL b1
for b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of NAT
      st b4 in Seg b1 & b2 = b3
   holds Proj(b3,b4) <= |.b2.| &
    (Proj(b3,b4)) ^2 <= |.b2.| ^2;

:: JORDAN2B:th 16
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of NAT
      st b3 = {b5 where b5 is Element of the carrier of TOP-REAL b1: Proj(b5,b4) < b2} &
         b4 in Seg b1
   holds b3 is open(TOP-REAL b1);

:: JORDAN2B:th 17
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of NAT
      st b3 = {b5 where b5 is Element of the carrier of TOP-REAL b1: b2 < Proj(b5,b4)} &
         b4 in Seg b1
   holds b3 is open(TOP-REAL b1);

:: JORDAN2B:th 18
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being real set
for b5 being Element of NAT
      st b2 = {b6 where b6 is Element of the carrier of TOP-REAL b1: b3 < Proj(b6,b5) & Proj(b6,b5) < b4} &
         b5 in Seg b1
   holds b2 is open(TOP-REAL b1);

:: JORDAN2B:th 19
theorem
for b1 being Element of NAT
for b2, b3 being real set
for b4 being Function-like quasi_total Relation of the carrier of TOP-REAL b1,the carrier of R^1
for b5 being Element of NAT
      st for b6 being Element of the carrier of TOP-REAL b1 holds
           b4 . b6 = Proj(b6,b5)
   holds b4 " {b6 where b6 is Element of REAL: b2 < b6 & b6 < b3} = {b6 where b6 is Element of the carrier of TOP-REAL b1: b2 < Proj(b6,b5) & Proj(b6,b5) < b3};

:: JORDAN2B:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Reflexive discerning symmetric triangle MetrStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of TopSpaceMetr b2
      st for b4 being real set
        for b5 being Element of the carrier of b2
        for b6 being Element of bool the carrier of TopSpaceMetr b2
              st 0 < b4 & b6 = Ball(b5,b4)
           holds b3 " b6 is open(b1)
   holds b3 is continuous(b1, TopSpaceMetr b2);

:: JORDAN2B:th 21
theorem
for b1 being Element of the carrier of RealSpace
for b2, b3 being real set
      st b3 = b1 & 0 < b2
   holds Ball(b1,b2) = {b4 where b4 is Element of REAL: b3 - b2 < b4 & b4 < b3 + b2};

:: JORDAN2B:th 22
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of the carrier of TOP-REAL b1,the carrier of R^1
for b3 being Element of NAT
      st b3 in Seg b1 &
         (for b4 being Element of the carrier of TOP-REAL b1 holds
            b2 . b4 = Proj(b4,b3))
   holds b2 is continuous(TOP-REAL b1, R^1);

:: JORDAN2B:th 23
theorem
for b1 being Element of REAL holds
   abs <*b1*> = <*abs b1*>;

:: JORDAN2B:th 24
theorem
for b1 being Element of the carrier of TOP-REAL 1 holds
   ex b2 being Element of REAL st
      b1 = <*b2*>;

:: JORDAN2B:funcnot 2 => FINSEQ_1:func 5
notation
  let a1 be real set;
  synonym |[a1]| for <*a1*>;
end;

:: JORDAN2B:funcnot 3 => JORDAN2B:func 2
definition
  let a1 be real set;
  redefine func |[a1]| -> Element of the carrier of TOP-REAL 1;
end;

:: JORDAN2B:th 26
theorem
for b1 being real set
for b2 being Element of REAL holds
   b2 * |[b1]| = |[b2 * b1]|;

:: JORDAN2B:th 27
theorem
for b1, b2 being real set holds
|[b1 + b2]| = |[b1]| + |[b2]|;

:: JORDAN2B:th 29
theorem
for b1, b2 being real set
      st |[b1]| = |[b2]|
   holds b1 = b2;

:: JORDAN2B:th 30
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
      st b1 = {b3 where b3 is Element of REAL: b3 < b2}
   holds b1 is open(R^1);

:: JORDAN2B:th 31
theorem
for b1 being Element of bool the carrier of R^1
for b2 being real set
      st b1 = {b3 where b3 is Element of REAL: b2 < b3}
   holds b1 is open(R^1);

:: JORDAN2B:th 32
theorem
for b1 being Element of bool the carrier of R^1
for b2, b3 being real set
      st b1 = {b4 where b4 is Element of REAL: b2 < b4 & b4 < b3}
   holds b1 is open(R^1);

:: JORDAN2B:th 33
theorem
for b1 being Element of the carrier of Euclid 1
for b2, b3 being real set
      st <*b3*> = b1 & 0 < b2
   holds Ball(b1,b2) = {<*b4*> where b4 is Element of REAL: b3 - b2 < b4 & b4 < b3 + b2};

:: JORDAN2B:th 34
theorem
for b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 1,the carrier of R^1
      st for b2 being Element of the carrier of TOP-REAL 1 holds
           b1 . b2 = Proj(b2,1)
   holds b1 is being_homeomorphism(TOP-REAL 1, R^1);

:: JORDAN2B:th 35
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   Proj(b1,1) = b1 `1 & Proj(b1,2) = b1 `2;

:: JORDAN2B:th 36
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   Proj(b1,1) = proj1 . b1 & Proj(b1,2) = proj2 . b1;