Article DIRORT, MML version 4.99.1005

:: DIRORT:prednot 1 => ANALOAF:pred 2
notation
  let a1 be non empty AffinStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  synonym a2,a3 '//' a4,a5 for a2,a3 // a4,a5;
end;

:: DIRORT:th 1
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 Gen b2,b3
   holds (for b4, b5, b6, b7, b8 being Element of the carrier of CESpace(b1,b2,b3) holds
    b4,b4 // b6,b8 &
     b4,b6 // b8,b8 &
     (b4,b6 // b5,b7 & b4,b6 // b7,b5 & b4 <> b6 implies b5 = b7) &
     (b4,b6 // b5,b7 & b4,b6 // b5,b8 & not b4,b6 // b7,b8 implies b4,b6 // b8,b7) &
     (b4,b6 // b5,b7 implies b6,b4 // b7,b5) &
     (b4,b6 // b5,b7 & b4,b6 // b7,b8 implies b4,b6 // b5,b8) &
     (b4,b5 // b6,b7 & not b6,b7 // b4,b5 implies b6,b7 // b5,b4)) &
    (for b4, b5, b6 being Element of the carrier of CESpace(b1,b2,b3) holds
    ex b7 being Element of the carrier of CESpace(b1,b2,b3) st
       b6 <> b7 & b6,b7 // b4,b5) &
    (for b4, b5, b6 being Element of the carrier of CESpace(b1,b2,b3) holds
    ex b7 being Element of the carrier of CESpace(b1,b2,b3) st
       b6 <> b7 & b4,b5 // b6,b7);

:: DIRORT:th 2
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 Gen b2,b3
   holds (for b4, b5, b6, b7, b8 being Element of the carrier of CMSpace(b1,b2,b3) holds
    b4,b4 // b6,b8 &
     b4,b6 // b8,b8 &
     (b4,b6 // b5,b7 & b4,b6 // b7,b5 & b4 <> b6 implies b5 = b7) &
     (b4,b6 // b5,b7 & b4,b6 // b5,b8 & not b4,b6 // b7,b8 implies b4,b6 // b8,b7) &
     (b4,b6 // b5,b7 implies b6,b4 // b7,b5) &
     (b4,b6 // b5,b7 & b4,b6 // b7,b8 implies b4,b6 // b5,b8) &
     (b4,b5 // b6,b7 & not b6,b7 // b4,b5 implies b6,b7 // b5,b4)) &
    (for b4, b5, b6 being Element of the carrier of CMSpace(b1,b2,b3) holds
    ex b7 being Element of the carrier of CMSpace(b1,b2,b3) st
       b6 <> b7 & b6,b7 // b4,b5) &
    (for b4, b5, b6 being Element of the carrier of CMSpace(b1,b2,b3) holds
    ex b7 being Element of the carrier of CMSpace(b1,b2,b3) st
       b6 <> b7 & b4,b5 // b6,b7);

:: DIRORT:attrnot 1 => DIRORT:attr 1
definition
  let a1 be non empty AffinStruct;
  attr a1 is Oriented_Orthogonality_Space-like means
    (for b1, b2, b3, b4, b5 being Element of the carrier of a1 holds
     b1,b1 // b3,b5 &
      b1,b3 // b5,b5 &
      (b1,b3 // b2,b4 & b1,b3 // b4,b2 & b1 <> b3 implies b2 = b4) &
      (b1,b3 // b2,b4 & b1,b3 // b2,b5 & not b1,b3 // b4,b5 implies b1,b3 // b5,b4) &
      (b1,b3 // b2,b4 implies b3,b1 // b4,b2) &
      (b1,b3 // b2,b4 & b1,b3 // b4,b5 implies b1,b3 // b2,b5) &
      (b1,b2 // b3,b4 & not b3,b4 // b1,b2 implies b3,b4 // b2,b1)) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b3 <> b4 & b3,b4 // b1,b2) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b3 <> b4 & b1,b2 // b3,b4);
end;

:: DIRORT:dfs 1
definiens
  let a1 be non empty AffinStruct;
To prove
     a1 is Oriented_Orthogonality_Space-like
it is sufficient to prove
  thus (for b1, b2, b3, b4, b5 being Element of the carrier of a1 holds
     b1,b1 // b3,b5 &
      b1,b3 // b5,b5 &
      (b1,b3 // b2,b4 & b1,b3 // b4,b2 & b1 <> b3 implies b2 = b4) &
      (b1,b3 // b2,b4 & b1,b3 // b2,b5 & not b1,b3 // b4,b5 implies b1,b3 // b5,b4) &
      (b1,b3 // b2,b4 implies b3,b1 // b4,b2) &
      (b1,b3 // b2,b4 & b1,b3 // b4,b5 implies b1,b3 // b2,b5) &
      (b1,b2 // b3,b4 & not b3,b4 // b1,b2 implies b3,b4 // b2,b1)) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b3 <> b4 & b3,b4 // b1,b2) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b3 <> b4 & b1,b2 // b3,b4);

:: DIRORT:def 1
theorem
for b1 being non empty AffinStruct holds
      b1 is Oriented_Orthogonality_Space-like
   iff
      (for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
       b2,b2 // b4,b6 &
        b2,b4 // b6,b6 &
        (b2,b4 // b3,b5 & b2,b4 // b5,b3 & b2 <> b4 implies b3 = b5) &
        (b2,b4 // b3,b5 & b2,b4 // b3,b6 & not b2,b4 // b5,b6 implies b2,b4 // b6,b5) &
        (b2,b4 // b3,b5 implies b4,b2 // b5,b3) &
        (b2,b4 // b3,b5 & b2,b4 // b5,b6 implies b2,b4 // b3,b6) &
        (b2,b3 // b4,b5 & not b4,b5 // b2,b3 implies b4,b5 // b3,b2)) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       ex b5 being Element of the carrier of b1 st
          b4 <> b5 & b4,b5 // b2,b3) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       ex b5 being Element of the carrier of b1 st
          b4 <> b5 & b2,b3 // b4,b5);

:: DIRORT:exreg 1
registration
  cluster non empty Oriented_Orthogonality_Space-like AffinStruct;
end;

:: DIRORT:modenot 1
definition
  mode Oriented_Orthogonality_Space is non empty Oriented_Orthogonality_Space-like AffinStruct;
end;

:: DIRORT:th 4
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 Gen b2,b3
   holds CMSpace(b1,b2,b3) is non empty Oriented_Orthogonality_Space-like AffinStruct;

:: DIRORT:th 5
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 Gen b2,b3
   holds CESpace(b1,b2,b3) is non empty Oriented_Orthogonality_Space-like AffinStruct;

:: DIRORT:th 6
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
   b5,b4 // b2,b3 & b5 <> b4;

:: DIRORT:th 8
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
   b2 <> b5 & (b3,b4 // b2,b5 or b3,b4 // b5,b2);

:: DIRORT:prednot 2 => DIRORT:pred 1
definition
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred A2,A3 _|_ A4,A5 means
    (not a2,a3 // a4,a5) implies a2,a3 // a5,a4;
end;

:: DIRORT:dfs 2
definiens
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     a2,a3 _|_ a4,a5
it is sufficient to prove
  thus (not a2,a3 // a4,a5) implies a2,a3 // a5,a4;

:: DIRORT:def 2
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b2,b3 _|_ b4,b5
iff
   (b2,b3 // b4,b5 or b2,b3 // b5,b4);

:: DIRORT:prednot 3 => DIRORT:pred 2
definition
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred A2,A3 // A4,A5 means
    ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & b1,b2 // a2,a3 & b1,b2 // a4,a5;
end;

:: DIRORT:dfs 3
definiens
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     a2,a3 // a4,a5
it is sufficient to prove
  thus ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & b1,b2 // a2,a3 & b1,b2 // a4,a5;

:: DIRORT:def 3
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b2,b3 // b4,b5
iff
   ex b6, b7 being Element of the carrier of b1 st
      b6 <> b7 & b6,b7 // b2,b3 & b6,b7 // b4,b5;

:: DIRORT:attrnot 2 => DIRORT:attr 2
definition
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
  attr a1 is bach_transitive means
    for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
          st b1,b2 // b4,b5 & b7,b8 // b4,b5 & b7,b8 // b3,b6 & b7 <> b8 & b4 <> b5
       holds b1,b2 // b3,b6;
end;

:: DIRORT:dfs 4
definiens
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
To prove
     a1 is bach_transitive
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
          st b1,b2 // b4,b5 & b7,b8 // b4,b5 & b7,b8 // b3,b6 & b7 <> b8 & b4 <> b5
       holds b1,b2 // b3,b6;

:: DIRORT:def 4
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct holds
      b1 is bach_transitive
   iff
      for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
            st b2,b3 // b5,b6 & b8,b9 // b5,b6 & b8,b9 // b4,b7 & b8 <> b9 & b5 <> b6
         holds b2,b3 // b4,b7;

:: DIRORT:attrnot 3 => DIRORT:attr 3
definition
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
  attr a1 is right_transitive means
    for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
          st b1,b2 // b4,b5 & b4,b5 // b7,b8 & b3,b6 // b7,b8 & b7 <> b8 & b4 <> b5
       holds b1,b2 // b3,b6;
end;

:: DIRORT:dfs 5
definiens
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
To prove
     a1 is right_transitive
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
          st b1,b2 // b4,b5 & b4,b5 // b7,b8 & b3,b6 // b7,b8 & b7 <> b8 & b4 <> b5
       holds b1,b2 // b3,b6;

:: DIRORT:def 5
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct holds
      b1 is right_transitive
   iff
      for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
            st b2,b3 // b5,b6 & b5,b6 // b8,b9 & b4,b7 // b8,b9 & b8 <> b9 & b5 <> b6
         holds b2,b3 // b4,b7;

:: DIRORT:attrnot 4 => DIRORT:attr 4
definition
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
  attr a1 is left_transitive means
    for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
          st b1,b2 // b4,b5 & b4,b5 // b7,b8 & b1,b2 // b3,b6 & b1 <> b2 & b4 <> b5
       holds b3,b6 // b7,b8;
end;

:: DIRORT:dfs 6
definiens
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
To prove
     a1 is left_transitive
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
          st b1,b2 // b4,b5 & b4,b5 // b7,b8 & b1,b2 // b3,b6 & b1 <> b2 & b4 <> b5
       holds b3,b6 // b7,b8;

:: DIRORT:def 6
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct holds
      b1 is left_transitive
   iff
      for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
            st b2,b3 // b5,b6 & b5,b6 // b8,b9 & b2,b3 // b4,b7 & b2 <> b3 & b5 <> b6
         holds b4,b7 // b8,b9;

:: DIRORT:attrnot 5 => DIRORT:attr 5
definition
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
  attr a1 is Euclidean_like means
    for b1, b2, b3, b4 being Element of the carrier of a1
          st b1,b2 // b3,b4
       holds b3,b4 // b2,b1;
end;

:: DIRORT:dfs 7
definiens
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
To prove
     a1 is Euclidean_like
it is sufficient to prove
  thus for b1, b2, b3, b4 being Element of the carrier of a1
          st b1,b2 // b3,b4
       holds b3,b4 // b2,b1;

:: DIRORT:def 7
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct holds
      b1 is Euclidean_like
   iff
      for b2, b3, b4, b5 being Element of the carrier of b1
            st b2,b3 // b4,b5
         holds b4,b5 // b3,b2;

:: DIRORT:attrnot 6 => DIRORT:attr 6
definition
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
  attr a1 is Minkowskian_like means
    for b1, b2, b3, b4 being Element of the carrier of a1
          st b1,b2 // b3,b4
       holds b3,b4 // b1,b2;
end;

:: DIRORT:dfs 8
definiens
  let a1 be non empty Oriented_Orthogonality_Space-like AffinStruct;
To prove
     a1 is Minkowskian_like
it is sufficient to prove
  thus for b1, b2, b3, b4 being Element of the carrier of a1
          st b1,b2 // b3,b4
       holds b3,b4 // b1,b2;

:: DIRORT:def 8
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct holds
      b1 is Minkowskian_like
   iff
      for b2, b3, b4, b5 being Element of the carrier of b1
            st b2,b3 // b4,b5
         holds b4,b5 // b2,b3;

:: DIRORT:th 9
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3 // b4,b4 & b4,b4 // b2,b3;

:: DIRORT:th 10
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b4,b5 // b2,b3;

:: DIRORT:th 11
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b3,b2 // b5,b4;

:: DIRORT:th 12
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct holds
      b1 is left_transitive
   iff
      for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
            st b2,b3 // b6,b7 & b2,b3 // b4,b5 & b2 <> b3
         holds b6,b7 // b4,b5;

:: DIRORT:th 13
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct holds
      b1 is bach_transitive
   iff
      for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
            st b2,b3 // b5,b6 & b5,b6 // b4,b7 & b5 <> b6
         holds b2,b3 // b4,b7;

:: DIRORT:th 14
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
   st b1 is bach_transitive
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2,b3 // b4,b5 & b4,b5 // b6,b7 & b4 <> b5
   holds b2,b3 // b6,b7;

:: DIRORT:th 15
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 non empty Oriented_Orthogonality_Space-like AffinStruct
      st Gen b2,b3 & b4 = CESpace(b1,b2,b3)
   holds b4 is Euclidean_like & b4 is left_transitive & b4 is right_transitive & b4 is bach_transitive;

:: DIRORT:exreg 2
registration
  cluster non empty Oriented_Orthogonality_Space-like bach_transitive right_transitive left_transitive Euclidean_like AffinStruct;
end;

:: DIRORT:th 16
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 non empty Oriented_Orthogonality_Space-like AffinStruct
      st Gen b2,b3 & b4 = CMSpace(b1,b2,b3)
   holds b4 is Minkowskian_like & b4 is left_transitive & b4 is right_transitive & b4 is bach_transitive;

:: DIRORT:exreg 3
registration
  cluster non empty Oriented_Orthogonality_Space-like bach_transitive right_transitive left_transitive Minkowskian_like AffinStruct;
end;

:: DIRORT:th 17
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
      st b1 is left_transitive
   holds b1 is right_transitive;

:: DIRORT:th 18
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
      st b1 is left_transitive
   holds b1 is bach_transitive;

:: DIRORT:th 19
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
      st b1 is bach_transitive
   holds    b1 is right_transitive
   iff
      for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
            st b2,b3 // b6,b7 & b4,b5 // b6,b7 & b6 <> b7
         holds b2,b3 // b4,b5;

:: DIRORT:th 20
theorem
for b1 being non empty Oriented_Orthogonality_Space-like AffinStruct
      st b1 is right_transitive & (b1 is Euclidean_like or b1 is Minkowskian_like)
   holds b1 is left_transitive;