Article PASCH, MML version 4.99.1005

:: PASCH:attrnot 1 => PASCH:attr 1
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  attr a1 is satisfying_Int_Par_Pasch means
    for b1, b2, b3, b4, b5 being Element of the carrier of a1
          st not LIN b5,b2,b3 & Mid b2,b5,b1 & LIN b5,b3,b4 & b2,b3 '||' b4,b1
       holds Mid b3,b5,b4;
end;

:: PASCH:dfs 1
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
     a1 is satisfying_Int_Par_Pasch
it is sufficient to prove
  thus for b1, b2, b3, b4, b5 being Element of the carrier of a1
          st not LIN b5,b2,b3 & Mid b2,b5,b1 & LIN b5,b3,b4 & b2,b3 '||' b4,b1
       holds Mid b3,b5,b4;

:: PASCH:def 1
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
      b1 is satisfying_Int_Par_Pasch
   iff
      for b2, b3, b4, b5, b6 being Element of the carrier of b1
            st not LIN b6,b3,b4 & Mid b3,b6,b2 & LIN b6,b4,b5 & b3,b4 '||' b5,b2
         holds Mid b4,b6,b5;

:: PASCH:prednot 1 => PASCH:attr 1
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  synonym a1 satisfies_Int_Par_Pasch for satisfying_Int_Par_Pasch;
end;

:: PASCH:attrnot 2 => PASCH:attr 2
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  attr a1 is satisfying_Ext_Par_Pasch means
    for b1, b2, b3, b4, b5 being Element of the carrier of a1
          st Mid b5,b2,b3 & LIN b5,b1,b4 & b1,b2 '||' b3,b4 & not LIN b5,b1,b2
       holds Mid b5,b1,b4;
end;

:: PASCH:dfs 2
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
     a1 is satisfying_Ext_Par_Pasch
it is sufficient to prove
  thus for b1, b2, b3, b4, b5 being Element of the carrier of a1
          st Mid b5,b2,b3 & LIN b5,b1,b4 & b1,b2 '||' b3,b4 & not LIN b5,b1,b2
       holds Mid b5,b1,b4;

:: PASCH:def 2
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
      b1 is satisfying_Ext_Par_Pasch
   iff
      for b2, b3, b4, b5, b6 being Element of the carrier of b1
            st Mid b6,b3,b4 & LIN b6,b2,b5 & b2,b3 '||' b4,b5 & not LIN b6,b2,b3
         holds Mid b6,b2,b5;

:: PASCH:prednot 2 => PASCH:attr 2
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  synonym a1 satisfies_Ext_Par_Pasch for satisfying_Ext_Par_Pasch;
end;

:: PASCH:attrnot 3 => PASCH:attr 3
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  attr a1 is satisfying_Gen_Par_Pasch means
    for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
          st not LIN b1,b2,b4 & b1,b4 '||' b2,b5 & b1,b4 '||' b3,b6 & Mid b1,b2,b3 & LIN b4,b5,b6
       holds Mid b4,b5,b6;
end;

:: PASCH:dfs 3
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
     a1 is satisfying_Gen_Par_Pasch
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
          st not LIN b1,b2,b4 & b1,b4 '||' b2,b5 & b1,b4 '||' b3,b6 & Mid b1,b2,b3 & LIN b4,b5,b6
       holds Mid b4,b5,b6;

:: PASCH:def 3
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
      b1 is satisfying_Gen_Par_Pasch
   iff
      for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
            st not LIN b2,b3,b5 & b2,b5 '||' b3,b6 & b2,b5 '||' b4,b7 & Mid b2,b3,b4 & LIN b5,b6,b7
         holds Mid b5,b6,b7;

:: PASCH:prednot 3 => PASCH:attr 3
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  synonym a1 satisfies_Gen_Par_Pasch for satisfying_Gen_Par_Pasch;
end;

:: PASCH:attrnot 4 => PASCH:attr 4
definition
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  attr a1 is satisfying_Ext_Bet_Pasch means
    for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
          st Mid b1,b2,b4 & Mid b2,b5,b3 & not LIN b1,b2,b3
       holds ex b7 being Element of the carrier of a1 st
          Mid b1,b7,b3 & Mid b7,b5,b4;
end;

:: PASCH:dfs 4
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
     a1 is satisfying_Ext_Bet_Pasch
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
          st Mid b1,b2,b4 & Mid b2,b5,b3 & not LIN b1,b2,b3
       holds ex b7 being Element of the carrier of a1 st
          Mid b1,b7,b3 & Mid b7,b5,b4;

:: PASCH:def 4
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
      b1 is satisfying_Ext_Bet_Pasch
   iff
      for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
            st Mid b2,b3,b5 & Mid b3,b6,b4 & not LIN b2,b3,b4
         holds ex b8 being Element of the carrier of b1 st
            Mid b2,b8,b4 & Mid b8,b6,b5;

:: PASCH:prednot 4 => PASCH:attr 4
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  synonym a1 satisfies_Ext_Bet_Pasch for satisfying_Ext_Bet_Pasch;
end;

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

:: PASCH:dfs 5
definiens
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
     a1 is satisfying_Int_Bet_Pasch
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
          st Mid b1,b2,b4 & Mid b1,b5,b3 & not LIN b1,b2,b3
       holds ex b7 being Element of the carrier of a1 st
          Mid b2,b7,b3 & Mid b5,b7,b4;

:: PASCH:def 5
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
      b1 is satisfying_Int_Bet_Pasch
   iff
      for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
            st Mid b2,b3,b5 & Mid b2,b6,b4 & not LIN b2,b3,b4
         holds ex b8 being Element of the carrier of b1 st
            Mid b3,b8,b4 & Mid b6,b8,b5;

:: PASCH:prednot 5 => PASCH:attr 5
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  synonym a1 satisfies_Int_Bet_Pasch for satisfying_Int_Bet_Pasch;
end;

:: PASCH:attrnot 6 => PASCH:attr 6
definition
  let a1 be non empty non trivial OAffinSpace-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 & not LIN b1,b2,b3
       holds ex b5 being Element of the carrier of a1 st
          Mid b1,b5,b4 & Mid b2,b5,b3;
end;

:: PASCH:dfs 6
definiens
  let a1 be non empty non trivial OAffinSpace-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 & not LIN b1,b2,b3
       holds ex b5 being Element of the carrier of a1 st
          Mid b1,b5,b4 & Mid b2,b5,b3;

:: PASCH:def 6
theorem
for b1 being non empty non trivial OAffinSpace-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 & not LIN b2,b3,b4
         holds ex b6 being Element of the carrier of b1 st
            Mid b2,b6,b5 & Mid b3,b6,b4;

:: PASCH:prednot 6 => PASCH:attr 6
notation
  let a1 be non empty non trivial OAffinSpace-like AffinStruct;
  synonym a1 satisfies_Fano for Fanoian;
end;

:: PASCH:th 7
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b3,b4 & b3 <> b4 & b2 <> b3
   holds ex b6 being Element of the carrier of b1 st
      b5,b3 // b3,b6 & b5,b2 '||' b4,b6 & b4 <> b6 & b3 <> b6;

:: PASCH:th 8
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b2,b4 & b2 <> b4 & b3 <> b2
   holds ex b6 being Element of the carrier of b1 st
      b2,b5 // b2,b6 & b5,b3 '||' b4,b6 & b4 <> b6;

:: PASCH:th 9
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 '||' b2,b4 & b2 <> b3
   holds ex b6 being Element of the carrier of b1 st
      b2,b5 '||' b2,b6 & b5,b3 '||' b4,b6;

:: PASCH:th 11
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st not LIN b2,b3,b4 & LIN b2,b4,b5 & LIN b2,b3,b6 & LIN b2,b3,b7 & b3,b4 '||' b5,b6 & b3,b4 '||' b5,b7
   holds b6 = b7;

:: PASCH: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 & b2,b3 '||' b4,b5 & b2,b3 '||' b4,b6 & b2,b4 '||' b3,b5 & b2,b4 '||' b3,b6
   holds b5 = b6;

:: PASCH: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 & Mid b3,b2,b5 & LIN b2,b4,b6 & b3,b4 '||' b6,b5
   holds Mid b4,b2,b6;

:: PASCH:th 14
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   b1 is satisfying_Int_Par_Pasch;

:: PASCH:th 15
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 Mid b2,b3,b4 & LIN b2,b5,b6 & b5,b3 '||' b4,b6 & not LIN b2,b5,b3
   holds Mid b2,b5,b6;

:: PASCH:th 16
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   b1 is satisfying_Ext_Par_Pasch;

:: PASCH:th 17
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st not LIN b2,b3,b4 & b2,b4 '||' b3,b5 & b2,b4 '||' b6,b7 & Mid b2,b3,b6 & LIN b4,b5,b7
   holds Mid b4,b5,b7;

:: PASCH:th 18
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   b1 is satisfying_Gen_Par_Pasch;

:: PASCH:th 19
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 & b4,b2 // b2,b6 & b3,b4 '||' b5,b6
   holds b3,b4 // b6,b5;

:: PASCH:th 20
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 & b2,b4 // b2,b6 & b3,b4 '||' b5,b6
   holds b3,b4 // b5,b6;

:: PASCH:th 21
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st not LIN b2,b3,b4 & b2,b3 '||' b4,b5 & b2,b4 '||' b3,b5
   holds b2,b3 // b4,b5 & b2,b4 // b3,b5;

:: PASCH:th 22
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 Mid b2,b3,b4 & b3,b5 // b4,b6 & b2,b5 // b2,b6 & not LIN b2,b6,b4 & b2 <> b3
   holds Mid b2,b5,b6;

:: PASCH:th 23
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 Mid b2,b3,b4 & b5,b3 // b6,b4 & b2,b5 // b2,b6 & not LIN b2,b4,b6 & b2 <> b5
   holds Mid b2,b5,b6;

:: PASCH:th 24
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,b4 // b2,b5 & b4,b3 // b5,b6 & LIN b3,b2,b6 & b2 <> b6
   holds not Mid b3,b2,b6;

:: PASCH:th 25
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b2,b4 & b3 <> b2
   holds ex b6 being Element of the carrier of b1 st
      b2,b5 // b2,b6 & b3,b5 // b4,b6;

:: PASCH:th 26
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st Mid b2,b3,b4
   holds ex b6 being Element of the carrier of b1 st
      Mid b2,b6,b5 & b4,b5 // b3,b6;

:: PASCH:th 27
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 <> b3 & Mid b2,b3,b4
   holds ex b6 being Element of the carrier of b1 st
      Mid b2,b5,b6 & b3,b5 // b4,b6;

:: PASCH:th 28
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st not LIN b2,b3,b4 & Mid b2,b5,b4
   holds ex b6 being Element of the carrier of b1 st
      Mid b2,b6,b3 & b3,b4 // b6,b5;

:: PASCH:th 29
theorem
for b1 being non empty non trivial OAffinSpace-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,b3 // b5,b4;

:: PASCH:th 30
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5 & not LIN b2,b3,b4
   holds ex b6 being Element of the carrier of b1 st
      Mid b2,b6,b5 & Mid b3,b6,b4;

:: PASCH:th 32
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   b1 is Fanoian;

:: PASCH:th 33
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st 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 b6,b2,b5 & LIN b6,b3,b4;

:: PASCH:th 34
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 b2,b3 '||' b4,b5 & b2,b4 '||' b3,b5 & not LIN b2,b3,b4 & LIN b6,b2,b5 & LIN b6,b3,b4
   holds not LIN b6,b2,b3;

:: PASCH:th 35
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 Mid b2,b3,b4 & Mid b3,b5,b6 & not LIN b2,b3,b6
   holds ex b7 being Element of the carrier of b1 st
      Mid b2,b7,b6 & Mid b7,b5,b4;

:: PASCH:th 36
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   b1 is satisfying_Ext_Bet_Pasch;

:: PASCH:th 37
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 Mid b2,b3,b4 & Mid b2,b5,b6 & not LIN b2,b3,b6
   holds ex b7 being Element of the carrier of b1 st
      Mid b3,b7,b6 & Mid b5,b7,b4;

:: PASCH:th 38
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
   b1 is satisfying_Int_Bet_Pasch;

:: PASCH:th 39
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st Mid b2,b3,b4 & b2,b3 // b5,b6 & not LIN b2,b3,b5 & LIN b5,b6,b7 & b2,b5 // b3,b6 & b2,b5 // b4,b7
   holds Mid b5,b6,b7;