Article TRANSLAC, MML version 4.99.1005

:: TRANSLAC: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 LIN b1,b2,b3;
end;

:: TRANSLAC: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 LIN b1,b2,b3;

:: TRANSLAC:def 1
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 LIN b2,b3,b4;

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

:: TRANSLAC:th 2
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
   st ex b2, b3, b4 being Element of the carrier of b1 st
        LIN b2,b3,b4 & b2 <> b3 & b2 <> b4 & b3 <> b4
for b2, b3 being Element of the carrier of b1
      st b2 <> b3
   holds ex b4 being Element of the carrier of b1 st
      LIN b2,b3,b4 & b2 <> b4 & b3 <> b4;

:: TRANSLAC:th 4
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b1 is Fanoian & b2,b3 // b4,b5 & b2,b4 // b3,b5 & not LIN b2,b3,b4
   holds ex b6 being Element of the carrier of b1 st
      LIN b3,b4,b6 & LIN b2,b5,b6;

:: TRANSLAC:th 5
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b5 is translation(b1) & not LIN b2,b5 . b2,b3 & b2,b5 . b2 // b3,b4 & b2,b3 // b5 . b2,b4
   holds b4 = b5 . b3;

:: TRANSLAC:th 6
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is translational
   iff
      for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
            st not LIN b2,b3,b4 & not LIN b2,b3,b5 & b2,b3 // b4,b6 & b2,b3 // b5,b7 & b2,b4 // b3,b6 & b2,b5 // b3,b7
         holds b4,b5 // b6,b7;

:: TRANSLAC:th 7
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Element of the carrier of b1 holds
   ex b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
      b3 is translation(b1) & b3 . b2 = b2;

:: TRANSLAC:th 8
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st (for b6, b7, b8 being Element of the carrier of b1
               st b6 <> b7 & LIN b6,b7,b8 & b8 <> b6
            holds b8 = b7) &
         b2,b3 // b4,b5 &
         b2,b4 // b3,b5 &
         not LIN b2,b3,b4
   holds b2,b5 // b3,b4;

:: TRANSLAC:th 9
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3 being Element of the carrier of b1
      st b1 is translational
   holds ex b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
      b4 is translation(b1) & b4 . b2 = b3;

:: TRANSLAC:th 10
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
      st for b2, b3 being Element of the carrier of b1 holds
        ex b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
           b4 is translation(b1) & b4 . b2 = b3
   holds b1 is translational;

:: TRANSLAC:th 11
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b3 is translation(b1) & b4 is translation(b1) & not LIN b2,b3 . b2,b4 . b2
   holds b3 * b4 = b4 * b3;

:: TRANSLAC:th 12
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b1 is translational & b2 is translation(b1) & b3 is translation(b1)
   holds b2 * b3 = b3 * b2;

:: TRANSLAC:th 13
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b3 is translation(b1) & b4 is translation(b1) & b2,b3 . b2 // b2,b4 . b2
   holds b2,b3 . b2 // b2,(b3 * b4) . b2;

:: TRANSLAC:th 14
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b1 is Fanoian & b1 is translational & b2 is translation(b1)
   holds ex b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
      b3 is translation(b1) & b3 * b3 = b2;

:: TRANSLAC:th 15
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b1 is Fanoian & b2 is translation(b1) & b2 * b2 = id the carrier of b1
   holds b2 = id the carrier of b1;

:: TRANSLAC:th 16
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b1 is translational & b1 is Fanoian & b2 is translation(b1) & b3 is translation(b1) & b4 is translation(b1) & b2 = b3 * b3 & b2 = b4 * b4
   holds b3 = b4;