Article PAPDESAF, MML version 4.99.1005

:: PAPDESAF:funcreg 1
registration
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  cluster Lambda a1 -> non trivial strict AffinSpace-like;
end;

:: PAPDESAF:funcreg 2
registration
  let a1 be non empty non trivial OAffinSpace-like 2-dimensional AffinStruct;
  cluster Lambda a1 -> strict 2-dimensional;
end;

:: PAPDESAF:th 2
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2 being set holds
   (b2 is Element of the carrier of b1 implies b2 is Element of the carrier of Lambda b1) &
    (b2 is Element of the carrier of Lambda b1 implies b2 is Element of the carrier of b1) &
    (b2 is Element of bool the carrier of b1 implies b2 is Element of bool the carrier of Lambda b1) &
    (b2 is Element of bool the carrier of Lambda b1 implies b2 is Element of bool the carrier of b1);

:: PAPDESAF:th 3
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5, b6, b7 being Element of the carrier of Lambda b1
      st b2 = b5 & b3 = b6 & b4 = b7
   holds    LIN b2,b3,b4
   iff
      LIN b5,b6,b7;

:: PAPDESAF:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being set holds
      b2 is Element of the carrier of OASpace b1
   iff
      b2 is Element of the carrier of b1;

:: PAPDESAF:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty non trivial OAffinSpace-like AffinStruct
   st b2 = OASpace b1
for b3, b4, b5, b6 being Element of the carrier of b2
for b7, b8, b9, b10 being Element of the carrier of b1
      st b3 = b7 & b4 = b8 & b5 = b9 & b6 = b10
   holds    b3,b4 '||' b5,b6
   iff
      b7,b8 '||' b9,b10;

:: PAPDESAF:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty non trivial OAffinSpace-like AffinStruct
      st b2 = OASpace b1
   holds ex b3, b4 being Element of the carrier of b1 st
      for b5, b6 being Element of REAL
            st (b5 * b3) + (b6 * b4) = 0. b1
         holds b5 = 0 & b6 = 0;

:: PAPDESAF:prednot 1 => AFF_2:attr 2
notation
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  synonym a1 satisfies_PAP' for Pappian;
end;

:: PAPDESAF:prednot 2 => AFF_2:attr 4
notation
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  synonym a1 satisfies_DES' for Desarguesian;
end;

:: PAPDESAF:prednot 3 => AFF_2:attr 7
notation
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  synonym a1 satisfies_TDES' for Moufangian;
end;

:: PAPDESAF:prednot 4 => AFF_2:attr 11
notation
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  synonym a1 satisfies_des' for translational;
end;

:: PAPDESAF:prednot 5 => TRANSLAC:attr 1
notation
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  synonym a1 satisfies_Fano for Fanoian;
end;

:: PAPDESAF:attrnot 1 => TRANSLAC:attr 1
definition
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
  attr a1 is Fanoian means
    for b1, b2, b3, b4 being Element of the carrier of a1
          st b1,b2 // b3,b4 & b1,b3 // b2,b4 & b1,b4 // b2,b3
       holds b1,b2 // b1,b3;
end;

:: PAPDESAF:dfs 1
definiens
  let a1 be non empty non trivial AffinSpace-like AffinStruct;
To prove
     a1 is Fanoian
it is sufficient to prove
  thus for b1, b2, b3, b4 being Element of the carrier of a1
          st b1,b2 // b3,b4 & b1,b3 // b2,b4 & b1,b4 // b2,b3
       holds b1,b2 // b1,b3;

:: PAPDESAF:def 5
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
      b1 is Fanoian
   iff
      for b2, b3, b4, b5 being Element of the carrier of b1
            st b2,b3 // b4,b5 & b2,b4 // b3,b5 & b2,b5 // b3,b4
         holds b2,b3 // b2,b4;

:: PAPDESAF:attrnot 2 => PAPDESAF:attr 1
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  attr a1 is Pappian means
    Lambda a1 is Pappian;
end;

:: PAPDESAF:dfs 2
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
     a1 is Pappian
it is sufficient to prove
  thus Lambda a1 is Pappian;

:: PAPDESAF:def 11
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
      b1 is Pappian
   iff
      Lambda b1 is Pappian;

:: PAPDESAF:attrnot 3 => PAPDESAF:attr 2
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  attr a1 is Desarguesian means
    Lambda a1 is Desarguesian;
end;

:: PAPDESAF:dfs 3
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
     a1 is Desarguesian
it is sufficient to prove
  thus Lambda a1 is Desarguesian;

:: PAPDESAF:def 12
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
      b1 is Desarguesian
   iff
      Lambda b1 is Desarguesian;

:: PAPDESAF:attrnot 4 => PAPDESAF:attr 3
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  attr a1 is Moufangian means
    Lambda a1 is Moufangian;
end;

:: PAPDESAF:dfs 4
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
     a1 is Moufangian
it is sufficient to prove
  thus Lambda a1 is Moufangian;

:: PAPDESAF:def 13
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
      b1 is Moufangian
   iff
      Lambda b1 is Moufangian;

:: PAPDESAF:attrnot 5 => PAPDESAF:attr 4
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  attr a1 is translation means
    Lambda a1 is translational;
end;

:: PAPDESAF:dfs 5
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
     a1 is translation
it is sufficient to prove
  thus Lambda a1 is translational;

:: PAPDESAF:def 14
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
      b1 is translation
   iff
      Lambda b1 is translational;

:: PAPDESAF:attrnot 6 => PAPDESAF:attr 5
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  attr a1 is satisfying_DES means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1,b2 // b1,b5 & b1,b3 // b1,b6 & b1,b4 // b1,b7 & not LIN b1,b2,b3 & not LIN b1,b2,b4 & b2,b3 // b5,b6 & b2,b4 // b5,b7
       holds b3,b4 // b6,b7;
end;

:: PAPDESAF:dfs 6
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
     a1 is satisfying_DES
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1,b2 // b1,b5 & b1,b3 // b1,b6 & b1,b4 // b1,b7 & not LIN b1,b2,b3 & not LIN b1,b2,b4 & b2,b3 // b5,b6 & b2,b4 // b5,b7
       holds b3,b4 // b6,b7;

:: PAPDESAF:def 15
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
      b1 is satisfying_DES
   iff
      for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
            st b2,b3 // b2,b6 & b2,b4 // b2,b7 & b2,b5 // b2,b8 & not LIN b2,b3,b4 & not LIN b2,b3,b5 & b3,b4 // b6,b7 & b3,b5 // b6,b8
         holds b4,b5 // b7,b8;

:: PAPDESAF:prednot 6 => PAPDESAF:attr 5
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  synonym a1 satisfies_DES for satisfying_DES;
end;

:: PAPDESAF:attrnot 7 => PAPDESAF:attr 6
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  attr a1 is satisfying_DES_1 means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b2,b1 // b1,b5 & b3,b1 // b1,b6 & b4,b1 // b1,b7 & not LIN b1,b2,b3 & not LIN b1,b2,b4 & b2,b3 // b6,b5 & b2,b4 // b7,b5
       holds b3,b4 // b7,b6;
end;

:: PAPDESAF:dfs 7
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
     a1 is satisfying_DES_1
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b2,b1 // b1,b5 & b3,b1 // b1,b6 & b4,b1 // b1,b7 & not LIN b1,b2,b3 & not LIN b1,b2,b4 & b2,b3 // b6,b5 & b2,b4 // b7,b5
       holds b3,b4 // b7,b6;

:: PAPDESAF:def 16
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
      b1 is satisfying_DES_1
   iff
      for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
            st b3,b2 // b2,b6 & b4,b2 // b2,b7 & b5,b2 // b2,b8 & not LIN b2,b3,b4 & not LIN b2,b3,b5 & b3,b4 // b7,b6 & b3,b5 // b8,b6
         holds b4,b5 // b8,b7;

:: PAPDESAF:prednot 7 => PAPDESAF:attr 6
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  synonym a1 satisfies_DES_1 for satisfying_DES_1;
end;

:: PAPDESAF:th 11
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
      st b1 is satisfying_DES_1
   holds b1 is satisfying_DES;

:: PAPDESAF:th 12
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not LIN b2,b3,b4 & b3,b2 // b2,b5 & LIN b2,b4,b6 & b3,b4 '||' b5,b6
   holds b4,b2 // b2,b6 & b3,b4 // b6,b5;

:: PAPDESAF:th 13
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not LIN b2,b3,b4 & b2,b3 // b2,b5 & LIN b2,b4,b6 & b3,b4 '||' b5,b6
   holds b2,b4 // b2,b6 & b3,b4 // b5,b6;

:: PAPDESAF:th 14
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
      st b1 is satisfying_DES_1
   holds Lambda b1 is Desarguesian;

:: PAPDESAF:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1
for b7 being Element of REAL
      st b2 - b3 = b7 * (b5 - b2) & b7 <> 0 & b2,b4 '||' b2,b6 & not b2,b3 '||' b2,b4 & b3,b4 '||' b5,b6
   holds b6 = b5 + ((- b7) " * (b4 - b3)) &
    b6 = b2 + ((- b7) " * (b4 - b2)) &
    b4 - b3 = (- b7) * (b6 - b5);

:: PAPDESAF:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty non trivial OAffinSpace-like AffinStruct
      st b2 = OASpace b1
   holds b2 is satisfying_DES_1;

:: PAPDESAF:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty non trivial OAffinSpace-like AffinStruct
      st b2 = OASpace b1
   holds b2 is satisfying_DES_1 & b2 is satisfying_DES;

:: PAPDESAF:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty non trivial OAffinSpace-like AffinStruct
      st b2 = OASpace b1
   holds Lambda b2 is Pappian;

:: PAPDESAF:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty non trivial OAffinSpace-like AffinStruct
      st b2 = OASpace b1
   holds Lambda b2 is Desarguesian;

:: PAPDESAF:th 21
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
      st b1 is Desarguesian
   holds b1 is Moufangian;

:: PAPDESAF:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty non trivial OAffinSpace-like AffinStruct
      st b2 = OASpace b1
   holds Lambda b2 is Moufangian;

:: PAPDESAF:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty non trivial OAffinSpace-like AffinStruct
      st b2 = OASpace b1
   holds Lambda b2 is translational;

:: PAPDESAF:th 24
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   Lambda b1 is Fanoian;

:: PAPDESAF:exreg 1
registration
  cluster non empty non trivial OAffinSpace-like Pappian Desarguesian Moufangian translation AffinStruct;
end;

:: PAPDESAF:exreg 2
registration
  cluster non empty non trivial strict AffinSpace-like 2-dimensional Pappian Desarguesian Moufangian translational Fanoian AffinStruct;
end;

:: PAPDESAF:exreg 3
registration
  cluster non empty non trivial strict AffinSpace-like Pappian Desarguesian Moufangian translational Fanoian AffinStruct;
end;

:: PAPDESAF:funcreg 3
registration
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  cluster Lambda a1 -> strict Fanoian;
end;

:: PAPDESAF:funcreg 4
registration
  let a1 be non empty non trivial OAffinSpace-like Pappian AffinStruct;
  cluster Lambda a1 -> strict Pappian;
end;

:: PAPDESAF:funcreg 5
registration
  let a1 be non empty non trivial OAffinSpace-like Desarguesian AffinStruct;
  cluster Lambda a1 -> strict Desarguesian;
end;

:: PAPDESAF:funcreg 6
registration
  let a1 be non empty non trivial OAffinSpace-like Moufangian AffinStruct;
  cluster Lambda a1 -> strict Moufangian;
end;

:: PAPDESAF:funcreg 7
registration
  let a1 be non empty non trivial OAffinSpace-like translation AffinStruct;
  cluster Lambda a1 -> strict translational;
end;