Article CIRCLED1, MML version 4.99.1005

:: CIRCLED1:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being circled Element of bool the carrier of b1 holds
b2 - b3 is circled(b1);

:: CIRCLED1:funcreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be circled Element of bool the carrier of a1;
  cluster a2 - a3 -> circled;
end;

:: CIRCLED1:attrnot 1 => RLTOPSP1:attr 3
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is circled means
    for b1 being Element of the carrier of a1
    for b2 being Element of REAL
          st abs b2 <= 1 & b1 in a2
       holds b2 * b1 in a2;
end;

:: CIRCLED1:dfs 1
definiens
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is circled
it is sufficient to prove
  thus for b1 being Element of the carrier of a1
    for b2 being Element of REAL
          st abs b2 <= 1 & b1 in a2
       holds b2 * b1 in a2;

:: CIRCLED1:def 1
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is circled(b1)
   iff
      for b3 being Element of the carrier of b1
      for b4 being Element of REAL
            st abs b4 <= 1 & b3 in b2
         holds b4 * b3 in b2;

:: CIRCLED1:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of REAL
      st b2 is circled(b1)
   holds b3 * b2 is circled(b1);

:: CIRCLED1:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of REAL
      st b2 is circled(b1) & b3 is circled(b1)
   holds (b4 * b2) + (b5 * b3) is circled(b1);

:: CIRCLED1:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of bool the carrier of b1
for b5, b6, b7 being Element of REAL
      st b2 is circled(b1) & b3 is circled(b1) & b4 is circled(b1)
   holds ((b5 * b2) + (b6 * b3)) + (b7 * b4) is circled(b1);

:: CIRCLED1:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   Up (0). b1 is circled(b1);

:: CIRCLED1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   Up (Omega). b1 is circled(b1);

:: CIRCLED1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being circled Element of bool the carrier of b1 holds
b2 /\ b3 is circled(b1);

:: CIRCLED1:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being circled Element of bool the carrier of b1 holds
b2 \/ b3 is circled(b1);

:: CIRCLED1:funcnot 1 => CIRCLED1:func 1
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  func Circled-Family A2 -> Element of bool bool the carrier of a1 means
    for b1 being Element of bool the carrier of a1 holds
          b1 in it
       iff
          b1 is circled(a1) & a2 c= b1;
end;

:: CIRCLED1:def 2
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1 holds
      b3 = Circled-Family b2
   iff
      for b4 being Element of bool the carrier of b1 holds
            b4 in b3
         iff
            b4 is circled(b1) & b2 c= b4;

:: CIRCLED1:funcnot 2 => CIRCLED1:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Element of bool the carrier of a1;
  func Cir A2 -> circled Element of bool the carrier of a1 equals
    meet Circled-Family a2;
end;

:: CIRCLED1:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1 holds
   Cir b2 = meet Circled-Family b2;

:: CIRCLED1:funcreg 2
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Element of bool the carrier of a1;
  cluster Circled-Family a2 -> non empty;
end;

:: CIRCLED1:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 c= b3
   holds Circled-Family b3 c= Circled-Family b2;

:: CIRCLED1:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 c= b3
   holds Cir b2 c= Cir b3;

:: CIRCLED1:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1 holds
   b2 c= Cir b2;

:: CIRCLED1:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being circled Element of bool the carrier of b1
      st b2 c= b3
   holds Cir b2 c= b3;

:: CIRCLED1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being circled Element of bool the carrier of b1 holds
   Cir b2 = b2;

:: CIRCLED1:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   Cir {} b1 = {};

:: CIRCLED1:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of REAL holds
   b3 * Cir b2 = Cir (b3 * b2);

:: CIRCLED1:attrnot 2 => CIRCLED1:attr 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Linear_Combination of a1;
  attr a2 is circled means
    ex b1 being FinSequence of the carrier of a1 st
       b1 is one-to-one &
        proj2 b1 = Carrier a2 &
        (ex b2 being FinSequence of REAL st
           len b2 = len b1 &
            Sum b2 = 1 &
            (for b3 being natural set
                  st b3 in dom b2
               holds b2 . b3 = a2 . (b1 . b3) & 0 <= b2 . b3));
end;

:: CIRCLED1:dfs 4
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Linear_Combination of a1;
To prove
     a2 is circled
it is sufficient to prove
  thus ex b1 being FinSequence of the carrier of a1 st
       b1 is one-to-one &
        proj2 b1 = Carrier a2 &
        (ex b2 being FinSequence of REAL st
           len b2 = len b1 &
            Sum b2 = 1 &
            (for b3 being natural set
                  st b3 in dom b2
               holds b2 . b3 = a2 . (b1 . b3) & 0 <= b2 . b3));

:: CIRCLED1:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
      b2 is circled(b1)
   iff
      ex b3 being FinSequence of the carrier of b1 st
         b3 is one-to-one &
          proj2 b3 = Carrier b2 &
          (ex b4 being FinSequence of REAL st
             len b4 = len b3 &
              Sum b4 = 1 &
              (for b5 being natural set
                    st b5 in dom b4
                 holds b4 . b5 = b2 . (b3 . b5) & 0 <= b4 . b5));

:: CIRCLED1:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
      st b2 is circled(b1)
   holds Carrier b2 <> {};

:: CIRCLED1:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
for b3 being Element of the carrier of b1
      st b2 is circled(b1) & b2 . b3 <= 0
   holds not b3 in Carrier b2;

:: CIRCLED1:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
      st b2 is circled(b1)
   holds b2 <> ZeroLC b1;

:: CIRCLED1:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   ex b2 being Linear_Combination of b1 st
      b2 is circled(b1);

:: CIRCLED1:exreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster Relation-like Function-like quasi_total complex-valued ext-real-valued real-valued circled Linear_Combination of a1;
end;

:: CIRCLED1:modenot 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  mode circled_Combination of a1 is circled Linear_Combination of a1;
end;

:: CIRCLED1:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Element of bool the carrier of b1 holds
   ex b3 being Linear_Combination of b2 st
      b3 is circled(b1);

:: CIRCLED1:exreg 2
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be non empty Element of bool the carrier of a1;
  cluster Relation-like Function-like quasi_total complex-valued ext-real-valued real-valued circled Linear_Combination of a2;
end;

:: CIRCLED1:modenot 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be non empty Element of bool the carrier of a1;
  mode circled_Combination of a2 is circled Linear_Combination of a2;
end;

:: CIRCLED1:funcnot 3 => CIRCLED1:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func circledComb A1 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 is circled Linear_Combination of a1;
end;

:: CIRCLED1:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being set holds
      b2 = circledComb b1
   iff
      for b3 being set holds
            b3 in b2
         iff
            b3 is circled Linear_Combination of b1;

:: CIRCLED1:funcnot 4 => CIRCLED1:func 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be non empty Element of bool the carrier of a1;
  func circledComb A2 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 is circled Linear_Combination of a2;
end;

:: CIRCLED1:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being set holds
      b3 = circledComb b2
   iff
      for b4 being set holds
            b4 in b3
         iff
            b4 is circled Linear_Combination of b2;

:: CIRCLED1:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
   ex b3 being circled Linear_Combination of b1 st
      Sum b3 = b2 &
       (for b4 being non empty Element of bool the carrier of b1
             st b2 in b4
          holds b3 is circled Linear_Combination of b4);

:: CIRCLED1:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
      st b2 <> b3
   holds ex b4 being circled Linear_Combination of b1 st
      for b5 being non empty Element of bool the carrier of b1
            st {b2,b3} c= b5
         holds b4 is circled Linear_Combination of b5;

:: CIRCLED1:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being circled Linear_Combination of b1
for b4, b5 being Element of REAL
      st 0 < b4 * b5
   holds Carrier ((b4 * b2) + (b5 * b3)) = (Carrier (b4 * b2)) \/ Carrier (b5 * b3);

:: CIRCLED1:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Linear_Combination of b1
      st b3 is circled(b1) & Carrier b3 = {b2}
   holds b3 . b2 = 1 & Sum b3 = (b3 . b2) * b2;

:: CIRCLED1:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Linear_Combination of b1
      st b4 is circled(b1) & Carrier b4 = {b2,b3} & b2 <> b3
   holds (b4 . b2) + (b4 . b3) = 1 &
    0 <= b4 . b2 &
    0 <= b4 . b3 &
    Sum b4 = ((b4 . b2) * b2) + ((b4 . b3) * b3);

:: CIRCLED1:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3 being Linear_Combination of {b2}
      st b3 is circled(b1)
   holds b3 . b2 = 1 & Sum b3 = (b3 . b2) * b2;

:: CIRCLED1:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Linear_Combination of {b2,b3}
      st b2 <> b3 & b4 is circled(b1)
   holds (b4 . b2) + (b4 . b3) = 1 &
    0 <= b4 . b2 &
    0 <= b4 . b3 &
    Sum b4 = ((b4 . b2) * b2) + ((b4 . b3) * b3);