Article CONMETR, MML version 4.99.1005

:: CONMETR:attrnot 1 => CONMETR:attr 1
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_OPAP means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
    for b8, b9 being Element of bool the carrier of a1
          st b1 in b8 & b2 in b8 & b3 in b8 & b4 in b8 & b1 in b9 & b5 in b9 & b6 in b9 & b7 in b9 & not b6 in b8 & not b4 in b9 & b8 _|_ b9 & b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & b4,b6 // b3,b5 & b4,b7 // b2,b5
       holds b2,b6 // b3,b7;
end;

:: CONMETR:dfs 1
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_OPAP
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
    for b8, b9 being Element of bool the carrier of a1
          st b1 in b8 & b2 in b8 & b3 in b8 & b4 in b8 & b1 in b9 & b5 in b9 & b6 in b9 & b7 in b9 & not b6 in b8 & not b4 in b9 & b8 _|_ b9 & b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & b4,b6 // b3,b5 & b4,b7 // b2,b5
       holds b2,b6 // b3,b7;

:: CONMETR:def 1
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_OPAP
   iff
      for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      for b9, b10 being Element of bool the carrier of b1
            st b2 in b9 & b3 in b9 & b4 in b9 & b5 in b9 & b2 in b10 & b6 in b10 & b7 in b10 & b8 in b10 & not b7 in b9 & not b5 in b10 & b9 _|_ b10 & b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b2 <> b7 & b2 <> b8 & b5,b7 // b4,b6 & b5,b8 // b3,b6
         holds b3,b7 // b4,b8;

:: CONMETR:attrnot 2 => CONMETR:attr 2
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_PAP means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
    for b8, b9 being Element of bool the carrier of a1
          st b8 is being_line(a1) & b9 is being_line(a1) & b1 in b8 & b2 in b8 & b3 in b8 & b4 in b8 & b1 in b9 & b5 in b9 & b6 in b9 & b7 in b9 & not b6 in b8 & not b4 in b9 & b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & b4,b6 // b3,b5 & b4,b7 // b2,b5
       holds b2,b6 // b3,b7;
end;

:: CONMETR:dfs 2
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_PAP
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
    for b8, b9 being Element of bool the carrier of a1
          st b8 is being_line(a1) & b9 is being_line(a1) & b1 in b8 & b2 in b8 & b3 in b8 & b4 in b8 & b1 in b9 & b5 in b9 & b6 in b9 & b7 in b9 & not b6 in b8 & not b4 in b9 & b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & b4,b6 // b3,b5 & b4,b7 // b2,b5
       holds b2,b6 // b3,b7;

:: CONMETR:def 2
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_PAP
   iff
      for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      for b9, b10 being Element of bool the carrier of b1
            st b9 is being_line(b1) & b10 is being_line(b1) & b2 in b9 & b3 in b9 & b4 in b9 & b5 in b9 & b2 in b10 & b6 in b10 & b7 in b10 & b8 in b10 & not b7 in b9 & not b5 in b10 & b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b2 <> b7 & b2 <> b8 & b5,b7 // b4,b6 & b5,b8 // b3,b6
         holds b3,b7 // b4,b8;

:: CONMETR:attrnot 3 => CONMETR:attr 3
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_MH1 means
    for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
    for b9, b10 being Element of bool the carrier of a1
          st b9 _|_ b10 & b1 in b9 & b3 in b9 & b5 in b9 & b7 in b9 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & not b2 in b9 & not b4 in b9 & b1,b2 _|_ b5,b6 & b2,b3 _|_ b6,b7 & b3,b4 _|_ b7,b8
       holds b1,b4 _|_ b5,b8;
end;

:: CONMETR:dfs 3
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_MH1
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
    for b9, b10 being Element of bool the carrier of a1
          st b9 _|_ b10 & b1 in b9 & b3 in b9 & b5 in b9 & b7 in b9 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & not b2 in b9 & not b4 in b9 & b1,b2 _|_ b5,b6 & b2,b3 _|_ b6,b7 & b3,b4 _|_ b7,b8
       holds b1,b4 _|_ b5,b8;

:: CONMETR:def 3
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_MH1
   iff
      for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
      for b10, b11 being Element of bool the carrier of b1
            st b10 _|_ b11 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & b3 in b11 & b5 in b11 & b7 in b11 & b9 in b11 & not b3 in b10 & not b5 in b10 & b2,b3 _|_ b6,b7 & b3,b4 _|_ b7,b8 & b4,b5 _|_ b8,b9
         holds b2,b5 _|_ b6,b9;

:: CONMETR:attrnot 4 => CONMETR:attr 4
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_MH2 means
    for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
    for b9, b10 being Element of bool the carrier of a1
          st b9 _|_ b10 & b1 in b9 & b3 in b9 & b6 in b9 & b8 in b9 & b2 in b10 & b4 in b10 & b5 in b10 & b7 in b10 & not b2 in b9 & not b4 in b9 & b1,b2 _|_ b5,b6 & b2,b3 _|_ b6,b7 & b3,b4 _|_ b7,b8
       holds b1,b4 _|_ b5,b8;
end;

:: CONMETR:dfs 4
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_MH2
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
    for b9, b10 being Element of bool the carrier of a1
          st b9 _|_ b10 & b1 in b9 & b3 in b9 & b6 in b9 & b8 in b9 & b2 in b10 & b4 in b10 & b5 in b10 & b7 in b10 & not b2 in b9 & not b4 in b9 & b1,b2 _|_ b5,b6 & b2,b3 _|_ b6,b7 & b3,b4 _|_ b7,b8
       holds b1,b4 _|_ b5,b8;

:: CONMETR:def 4
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_MH2
   iff
      for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
      for b10, b11 being Element of bool the carrier of b1
            st b10 _|_ b11 & b2 in b10 & b4 in b10 & b7 in b10 & b9 in b10 & b3 in b11 & b5 in b11 & b6 in b11 & b8 in b11 & not b3 in b10 & not b5 in b10 & b2,b3 _|_ b6,b7 & b3,b4 _|_ b7,b8 & b4,b5 _|_ b8,b9
         holds b2,b5 _|_ b6,b9;

:: CONMETR:attrnot 5 => CONMETR:attr 5
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_TDES means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & not LIN b4,b5,b2 & not LIN b4,b5,b6 & LIN b1,b2,b3 & LIN b1,b4,b5 & LIN b1,b6,b7 & b2,b4 // b3,b5 & b2,b4 // b1,b6 & b4,b6 // b5,b7
       holds b2,b6 // b3,b7;
end;

:: CONMETR:dfs 5
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_TDES
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 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & not LIN b4,b5,b2 & not LIN b4,b5,b6 & LIN b1,b2,b3 & LIN b1,b4,b5 & LIN b1,b6,b7 & b2,b4 // b3,b5 & b2,b4 // b1,b6 & b4,b6 // b5,b7
       holds b2,b6 // b3,b7;

:: CONMETR:def 5
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_TDES
   iff
      for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
            st b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b2 <> b7 & b2 <> b8 & not LIN b5,b6,b3 & not LIN b5,b6,b7 & LIN b2,b3,b4 & LIN b2,b5,b6 & LIN b2,b7,b8 & b3,b5 // b4,b6 & b3,b5 // b2,b7 & b5,b7 // b6,b8
         holds b3,b7 // b4,b8;

:: CONMETR:attrnot 6 => CONMETR:attr 6
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_SCH means
    for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
    for b9, b10 being Element of bool the carrier of a1
          st b9 is being_line(a1) & b10 is being_line(a1) & b1 in b9 & b3 in b9 & b5 in b9 & b7 in b9 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & not b4 in b9 & not b2 in b9 & not b6 in b9 & not b8 in b9 & not b1 in b10 & not b3 in b10 & not b5 in b10 & not b7 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
       holds b3,b4 // b7,b8;
end;

:: CONMETR:dfs 6
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_SCH
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
    for b9, b10 being Element of bool the carrier of a1
          st b9 is being_line(a1) & b10 is being_line(a1) & b1 in b9 & b3 in b9 & b5 in b9 & b7 in b9 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & not b4 in b9 & not b2 in b9 & not b6 in b9 & not b8 in b9 & not b1 in b10 & not b3 in b10 & not b5 in b10 & not b7 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
       holds b3,b4 // b7,b8;

:: CONMETR:def 6
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_SCH
   iff
      for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
      for b10, b11 being Element of bool the carrier of b1
            st b10 is being_line(b1) & b11 is being_line(b1) & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & b3 in b11 & b5 in b11 & b7 in b11 & b9 in b11 & not b5 in b10 & not b3 in b10 & not b7 in b10 & not b9 in b10 & not b2 in b11 & not b4 in b11 & not b6 in b11 & not b8 in b11 & b4,b3 // b8,b7 & b3,b2 // b7,b6 & b2,b5 // b6,b9
         holds b4,b5 // b8,b9;

:: CONMETR:attrnot 7 => CONMETR:attr 7
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_OSCH means
    for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
    for b9, b10 being Element of bool the carrier of a1
          st b9 _|_ b10 & b1 in b9 & b3 in b9 & b5 in b9 & b7 in b9 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & not b4 in b9 & not b2 in b9 & not b6 in b9 & not b8 in b9 & not b1 in b10 & not b3 in b10 & not b5 in b10 & not b7 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
       holds b3,b4 // b7,b8;
end;

:: CONMETR:dfs 7
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_OSCH
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1
    for b9, b10 being Element of bool the carrier of a1
          st b9 _|_ b10 & b1 in b9 & b3 in b9 & b5 in b9 & b7 in b9 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & not b4 in b9 & not b2 in b9 & not b6 in b9 & not b8 in b9 & not b1 in b10 & not b3 in b10 & not b5 in b10 & not b7 in b10 & b3,b2 // b7,b6 & b2,b1 // b6,b5 & b1,b4 // b5,b8
       holds b3,b4 // b7,b8;

:: CONMETR:def 7
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_OSCH
   iff
      for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
      for b10, b11 being Element of bool the carrier of b1
            st b10 _|_ b11 & b2 in b10 & b4 in b10 & b6 in b10 & b8 in b10 & b3 in b11 & b5 in b11 & b7 in b11 & b9 in b11 & not b5 in b10 & not b3 in b10 & not b7 in b10 & not b9 in b10 & not b2 in b11 & not b4 in b11 & not b6 in b11 & not b8 in b11 & b4,b3 // b8,b7 & b3,b2 // b7,b6 & b2,b5 // b6,b9
         holds b4,b5 // b8,b9;

:: CONMETR:attrnot 8 => CONMETR:attr 8
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_des means
    for b1, b2, b3, b4, b5, b6 being Element of the carrier of a1
          st not LIN b1,b2,b3 & not LIN b1,b2,b5 & b1,b2 // b3,b4 & b1,b2 // b5,b6 & b1,b3 // b2,b4 & b1,b5 // b2,b6
       holds b3,b5 // b4,b6;
end;

:: CONMETR:dfs 8
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_des
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,b3 & not LIN b1,b2,b5 & b1,b2 // b3,b4 & b1,b2 // b5,b6 & b1,b3 // b2,b4 & b1,b5 // b2,b6
       holds b3,b5 // b4,b6;

:: CONMETR:def 8
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_des
   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,b6 & b2,b3 // b4,b5 & b2,b3 // b6,b7 & b2,b4 // b3,b5 & b2,b6 // b3,b7
         holds b4,b6 // b5,b7;

:: CONMETR:prednot 1 => CONMETR:attr 1
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym OPAP_holds_in a1 for satisfying_OPAP;
end;

:: CONMETR:prednot 2 => CONMETR:attr 2
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym PAP_holds_in a1 for satisfying_PAP;
end;

:: CONMETR:prednot 3 => CONMETR:attr 3
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym MH1_holds_in a1 for satisfying_MH1;
end;

:: CONMETR:prednot 4 => CONMETR:attr 4
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym MH2_holds_in a1 for satisfying_MH2;
end;

:: CONMETR:prednot 5 => CONMETR:attr 5
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym TDES_holds_in a1 for satisfying_TDES;
end;

:: CONMETR:prednot 6 => CONMETR:attr 6
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym SCH_holds_in a1 for satisfying_SCH;
end;

:: CONMETR:prednot 7 => CONMETR:attr 7
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym OSCH_holds_in a1 for satisfying_OSCH;
end;

:: CONMETR:prednot 8 => CONMETR:attr 8
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym des_holds_in a1 for satisfying_des;
end;

:: CONMETR:th 1
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
   ex b2, b3, b4 being Element of the carrier of b1 st
      LIN b2,b3,b4 & b2 <> b3 & b3 <> b4 & b4 <> b2;

:: CONMETR:th 2
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
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;

:: CONMETR:th 3
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st b2 is being_line(b1)
   holds ex b4 being Element of bool the carrier of b1 st
      b3 in b4 & b2 _|_ b4;

:: CONMETR:th 4
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of bool the carrier of b1
      st b5 is being_line(b1) & b2 in b5 & b3 in b5 & b4 in b5
   holds LIN b2,b3,b4;

:: CONMETR:th 5
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
      st b4 is being_line(b1) & b5 is being_line(b1) & b2 in b4 & b3 in b4 & b2 in b5 & b3 in b5 & b2 <> b3
   holds b4 = b5;

:: CONMETR:th 6
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b1
for b7 being Element of bool the carrier of Af b1
for b8, b9 being Element of the carrier of Af b1
      st b4 = b8 & b5 = b9 & b6 = b7 & b2 in b6 & b3 in b6 & b8,b9 // b7
   holds b4,b5 // b2,b3;

:: CONMETR:th 7
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_TDES
   holds Af b1 is Moufangian;

:: CONMETR:th 8
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st Af b1 is translational
   holds b1 is satisfying_des;

:: CONMETR:th 9
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_MH1
   holds b1 is satisfying_OSCH;

:: CONMETR:th 10
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_MH2
   holds b1 is satisfying_OSCH;

:: CONMETR:th 11
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_AH
   holds b1 is satisfying_TDES;

:: CONMETR:th 12
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_OSCH & b1 is satisfying_TDES
   holds b1 is satisfying_SCH;

:: CONMETR:th 13
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_OPAP & b1 is satisfying_DES
   holds b1 is satisfying_PAP;

:: CONMETR:th 14
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_MH1 & b1 is satisfying_MH2
   holds b1 is satisfying_OPAP;

:: CONMETR:th 15
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_3H
   holds b1 is satisfying_OPAP;