Article ANALMETR, MML version 4.99.1005

:: ANALMETR:prednot 1 => ANALMETR:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Element of the carrier of a1;
  pred Gen A2,A3 means
    (for b1 being Element of the carrier of a1 holds
        ex b2, b3 being Element of REAL st
           b1 = (b2 * a2) + (b3 * a3)) &
     (for b1, b2 being Element of REAL
           st (b1 * a2) + (b2 * a3) = 0. a1
        holds b1 = 0 & b2 = 0);
end;

:: ANALMETR:dfs 1
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Element of the carrier of a1;
To prove
     Gen a2,a3
it is sufficient to prove
  thus (for b1 being Element of the carrier of a1 holds
        ex b2, b3 being Element of REAL st
           b1 = (b2 * a2) + (b3 * a3)) &
     (for b1, b2 being Element of REAL
           st (b1 * a2) + (b2 * a3) = 0. a1
        holds b1 = 0 & b2 = 0);

:: ANALMETR:def 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 holds
   Gen b2,b3
iff
   (for b4 being Element of the carrier of b1 holds
       ex b5, b6 being Element of REAL st
          b4 = (b5 * b2) + (b6 * b3)) &
    (for b4, b5 being Element of REAL
          st (b4 * b2) + (b5 * b3) = 0. b1
       holds b4 = 0 & b5 = 0);

:: ANALMETR:prednot 2 => ANALMETR:pred 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred A2,A3 are_Ort_wrt A4,A5 means
    ex b1, b2, b3, b4 being Element of REAL st
       a2 = (b1 * a4) + (b2 * a5) & a3 = (b3 * a4) + (b4 * a5) & (b1 * b3) + (b2 * b4) = 0;
end;

:: ANALMETR:dfs 2
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     a2,a3 are_Ort_wrt a4,a5
it is sufficient to prove
  thus ex b1, b2, b3, b4 being Element of REAL st
       a2 = (b1 * a4) + (b2 * a5) & a3 = (b3 * a4) + (b4 * a5) & (b1 * b3) + (b2 * b4) = 0;

:: ANALMETR:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b2,b3 are_Ort_wrt b4,b5
iff
   ex b6, b7, b8, b9 being Element of REAL st
      b2 = (b6 * b4) + (b7 * b5) & b3 = (b8 * b4) + (b9 * b5) & (b6 * b8) + (b7 * b9) = 0;

:: ANALMETR:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st Gen b4,b5
   holds    b2,b3 are_Ort_wrt b4,b5
   iff
      for b6, b7, b8, b9 being Element of REAL
            st b2 = (b6 * b4) + (b7 * b5) & b3 = (b8 * b4) + (b9 * b5)
         holds (b6 * b8) + (b7 * b9) = 0;

:: ANALMETR:th 6
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 holds
b2,b3 are_Ort_wrt b2,b3;

:: ANALMETR:th 7
theorem
ex b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct st
   ex b2, b3 being Element of the carrier of b1 st
      Gen b2,b3;

:: ANALMETR:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 are_Ort_wrt b4,b5
   holds b3,b2 are_Ort_wrt b4,b5;

:: ANALMETR:th 9
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
for b4, b5 being Element of the carrier of b1 holds
b4,0. b1 are_Ort_wrt b2,b3 & 0. b1,b5 are_Ort_wrt b2,b3;

:: ANALMETR:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7 being Element of REAL
      st b2,b3 are_Ort_wrt b4,b5
   holds b6 * b2,b7 * b3 are_Ort_wrt b4,b5;

:: ANALMETR:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7 being Element of REAL
      st b2,b3 are_Ort_wrt b4,b5
   holds b6 * b2,b3 are_Ort_wrt b4,b5 & b2,b7 * b3 are_Ort_wrt b4,b5;

:: ANALMETR:th 12
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
for b4 being Element of the carrier of b1 holds
   ex b5 being Element of the carrier of b1 st
      b4,b5 are_Ort_wrt b2,b3 & b5 <> 0. b1;

:: ANALMETR:th 13
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
      st Gen b2,b3 & b4,b5 are_Ort_wrt b2,b3 & b4,b6 are_Ort_wrt b2,b3 & b4 <> 0. b1
   holds ex b7, b8 being Element of REAL st
      b7 * b5 = b8 * b6 & (b7 = 0 implies b8 <> 0);

:: ANALMETR:th 14
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
      st Gen b2,b3 & b4,b5 are_Ort_wrt b2,b3 & b4,b6 are_Ort_wrt b2,b3
   holds b4,b5 + b6 are_Ort_wrt b2,b3 & b4,b5 - b6 are_Ort_wrt b2,b3;

:: ANALMETR:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st Gen b2,b3 & b4,b4 are_Ort_wrt b2,b3
   holds b4 = 0. b1;

:: ANALMETR:th 16
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
      st Gen b2,b3 & b4,b5 - b6 are_Ort_wrt b2,b3 & b5,b6 - b4 are_Ort_wrt b2,b3
   holds b6,b4 - b5 are_Ort_wrt b2,b3;

:: ANALMETR:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
      st Gen b2,b3 & b4 <> 0. b1
   holds ex b6 being Element of REAL st
      b5 - (b6 * b4),b4 are_Ort_wrt b2,b3;

:: ANALMETR:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   (b2,b3 // b4,b5 or b2,b3 // b5,b4)
iff
   ex b6, b7 being Element of REAL st
      b6 * (b3 - b2) = b7 * (b5 - b4) &
       (b6 = 0 implies b7 <> 0);

:: ANALMETR:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   [[b2,b3],[b4,b5]] in lambda DirPar b1
iff
   ex b6, b7 being Element of REAL st
      b6 * (b3 - b2) = b7 * (b5 - b4) &
       (b6 = 0 implies b7 <> 0);

:: ANALMETR:prednot 3 => ANALMETR:pred 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5, a6, a7 be Element of the carrier of a1;
  pred A2,A3,A4,A5 are_Ort_wrt A6,A7 means
    a3 - a2,a5 - a4 are_Ort_wrt a6,a7;
end;

:: ANALMETR:dfs 3
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4, a5, a6, a7 be Element of the carrier of a1;
To prove
     a2,a3,a4,a5 are_Ort_wrt a6,a7
it is sufficient to prove
  thus a3 - a2,a5 - a4 are_Ort_wrt a6,a7;

:: ANALMETR:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
   b2,b3,b4,b5 are_Ort_wrt b6,b7
iff
   b3 - b2,b5 - b4 are_Ort_wrt b6,b7;

:: ANALMETR:funcnot 1 => ANALMETR:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Element of the carrier of a1;
  func Orthogonality(A1,A2,A3) -> Relation of [:the carrier of a1,the carrier of a1:],[:the carrier of a1,the carrier of a1:] means
    for b1, b2 being set holds
       [b1,b2] in it
    iff
       ex b3, b4, b5, b6 being Element of the carrier of a1 st
          b1 = [b3,b4] & b2 = [b5,b6] & b3,b4,b5,b6 are_Ort_wrt a2,a3;
end;

:: ANALMETR:def 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
for b4 being Relation of [:the carrier of b1,the carrier of b1:],[:the carrier of b1,the carrier of b1:] holds
      b4 = Orthogonality(b1,b2,b3)
   iff
      for b5, b6 being set holds
         [b5,b6] in b4
      iff
         ex b7, b8, b9, b10 being Element of the carrier of b1 st
            b5 = [b7,b8] & b6 = [b9,b10] & b7,b8,b9,b10 are_Ort_wrt b2,b3;

:: ANALMETR:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   the carrier of Lambda OASpace b1 = the carrier of b1;

:: ANALMETR:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   the CONGR of Lambda OASpace b1 = lambda DirPar b1;

:: ANALMETR:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8, b9 being Element of the carrier of Lambda OASpace b1
      st b6 = b2 & b7 = b3 & b8 = b4 & b9 = b5
   holds    b6,b7 // b8,b9
   iff
      ex b10, b11 being Element of REAL st
         b10 * (b3 - b2) = b11 * (b5 - b4) &
          (b10 = 0 implies b11 <> 0);

:: ANALMETR:structnot 1 => ANALMETR:struct 1
definition
  struct(AffinStruct) ParOrtStr(#
    carrier -> set,
    CONGR -> Relation of [:the carrier of it,the carrier of it:],[:the carrier of it,the carrier of it:],
    orthogonality -> Relation of [:the carrier of it,the carrier of it:],[:the carrier of it,the carrier of it:]
  #);
end;

:: ANALMETR:attrnot 1 => ANALMETR:attr 1
definition
  let a1 be ParOrtStr;
  attr a1 is strict;
end;

:: ANALMETR:exreg 1
registration
  cluster strict ParOrtStr;
end;

:: ANALMETR:aggrnot 1 => ANALMETR:aggr 1
definition
  let a1 be set;
  let a2, a3 be Relation of [:a1,a1:],[:a1,a1:];
  aggr ParOrtStr(#a1,a2,a3#) -> strict ParOrtStr;
end;

:: ANALMETR:selnot 1 => ANALMETR:sel 1
definition
  let a1 be ParOrtStr;
  sel the orthogonality of a1 -> Relation of [:the carrier of a1,the carrier of a1:],[:the carrier of a1,the carrier of a1:];
end;

:: ANALMETR:exreg 2
registration
  cluster non empty ParOrtStr;
end;

:: ANALMETR:prednot 4 => ANALMETR:pred 4
definition
  let a1 be non empty ParOrtStr;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred A2,A3 _|_ A4,A5 means
    [[a2,a3],[a4,a5]] in the orthogonality of a1;
end;

:: ANALMETR:dfs 5
definiens
  let a1 be non empty ParOrtStr;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     a2,a3 _|_ a4,a5
it is sufficient to prove
  thus [[a2,a3],[a4,a5]] in the orthogonality of a1;

:: ANALMETR:def 6
theorem
for b1 being non empty ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b2,b3 _|_ b4,b5
iff
   [[b2,b3],[b4,b5]] in the orthogonality of b1;

:: ANALMETR:funcnot 2 => ANALMETR:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Element of the carrier of a1;
  func AMSpace(A1,A2,A3) -> strict ParOrtStr equals
    ParOrtStr(#the carrier of a1,lambda DirPar a1,Orthogonality(a1,a2,a3)#);
end;

:: ANALMETR:def 7
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 holds
AMSpace(b1,b2,b3) = ParOrtStr(#the carrier of b1,lambda DirPar b1,Orthogonality(b1,b2,b3)#);

:: ANALMETR:funcreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Element of the carrier of a1;
  cluster AMSpace(a1,a2,a3) -> non empty strict;
end;

:: ANALMETR:th 28
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 holds
the carrier of AMSpace(b1,b2,b3) = the carrier of b1 &
 the CONGR of AMSpace(b1,b2,b3) = lambda DirPar b1 &
 the orthogonality of AMSpace(b1,b2,b3) = Orthogonality(b1,b2,b3);

:: ANALMETR:funcnot 3 => ANALMETR:func 3
definition
  let a1 be non empty ParOrtStr;
  func Af A1 -> strict AffinStruct equals
    AffinStruct(#the carrier of a1,the CONGR of a1#);
end;

:: ANALMETR:def 8
theorem
for b1 being non empty ParOrtStr holds
   Af b1 = AffinStruct(#the carrier of b1,the CONGR of b1#);

:: ANALMETR:funcreg 2
registration
  let a1 be non empty ParOrtStr;
  cluster Af a1 -> non empty strict;
end;

:: ANALMETR:th 30
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 holds
Af AMSpace(b1,b2,b3) = Lambda OASpace b1;

:: ANALMETR:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
for b8, b9, b10, b11 being Element of the carrier of AMSpace(b1,b6,b7)
      st b8 = b2 & b9 = b3 & b10 = b4 & b11 = b5
   holds    b8,b10 _|_ b9,b11
   iff
      b2,b4,b3,b5 are_Ort_wrt b6,b7;

:: ANALMETR:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
for b8, b9, b10, b11 being Element of the carrier of AMSpace(b1,b2,b3)
      st b8 = b4 & b9 = b5 & b10 = b6 & b11 = b7
   holds    b8,b9 // b10,b11
   iff
      ex b12, b13 being Element of REAL st
         b12 * (b5 - b4) = b13 * (b7 - b6) &
          (b12 = 0 implies b13 <> 0);

:: ANALMETR:th 33
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, b5, b6, b7 being Element of the carrier of AMSpace(b1,b2,b3)
      st b4,b5 _|_ b6,b7
   holds b6,b7 _|_ b4,b5;

:: ANALMETR:th 34
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, b5, b6, b7 being Element of the carrier of AMSpace(b1,b2,b3)
      st b4,b5 _|_ b6,b7
   holds b4,b5 _|_ b7,b6;

:: ANALMETR:th 35
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
for b4, b5, b6 being Element of the carrier of AMSpace(b1,b2,b3) holds
b4,b5 _|_ b6,b6;

:: ANALMETR:th 36
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, b5, b6, b7, b8, b9 being Element of the carrier of AMSpace(b1,b2,b3)
      st b4,b5 _|_ b6,b7 & b4,b5 // b8,b9 & b4 <> b5
   holds b6,b7 _|_ b8,b9;

:: ANALMETR:th 37
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
for b4, b5, b6 being Element of the carrier of AMSpace(b1,b2,b3) holds
ex b7 being Element of the carrier of AMSpace(b1,b2,b3) st
   b4,b5 _|_ b6,b7 & b6 <> b7;

:: ANALMETR:th 38
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, b5, b6, b7, b8, b9 being Element of the carrier of AMSpace(b1,b2,b3)
      st Gen b2,b3 & b4,b5 _|_ b6,b7 & b4,b5 _|_ b8,b9 & b4 <> b5
   holds b6,b7 // b8,b9;

:: ANALMETR:th 39
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, b5, b6, b7, b8 being Element of the carrier of AMSpace(b1,b2,b3)
      st Gen b2,b3 & b4,b5 _|_ b6,b7 & b4,b5 _|_ b6,b8
   holds b4,b5 _|_ b7,b8;

:: ANALMETR:th 40
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, b5 being Element of the carrier of AMSpace(b1,b2,b3)
      st Gen b2,b3 & b4,b5 _|_ b4,b5
   holds b4 = b5;

:: ANALMETR:th 41
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, b5, b6, b7 being Element of the carrier of AMSpace(b1,b2,b3)
      st Gen b2,b3 & b4,b5 _|_ b6,b7 & b6,b5 _|_ b7,b4
   holds b7,b5 _|_ b4,b6;

:: ANALMETR:th 42
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, b5 being Element of the carrier of AMSpace(b1,b2,b3)
   st Gen b2,b3 & b4 <> b5
for b6 being Element of the carrier of AMSpace(b1,b2,b3) holds
   ex b7 being Element of the carrier of AMSpace(b1,b2,b3) st
      b4,b5 // b4,b7 & b4,b5 _|_ b7,b6;

:: ANALMETR:attrnot 2 => ANALMETR:attr 2
definition
  let a1 be non empty ParOrtStr;
  attr a1 is OrtAfSp-like means
    AffinStruct(#the carrier of a1,the CONGR of a1#) is non empty non trivial AffinSpace-like AffinStruct &
     (for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
     (b1,b2 _|_ b1,b2 implies b1 = b2) &
      b1,b2 _|_ b3,b3 &
      (b1,b2 _|_ b3,b4 implies b1,b2 _|_ b4,b3 & b3,b4 _|_ b1,b2) &
      (b1,b2 _|_ b5,b6 & b1,b2 // b7,b8 & not b5,b6 _|_ b7,b8 implies b1 = b2) &
      (b1,b2 _|_ b5,b6 & b1,b2 _|_ b5,b8 implies b1,b2 _|_ b6,b8)) &
     (for b1, b2, b3 being Element of the carrier of a1
           st b1 <> b2
        holds ex b4 being Element of the carrier of a1 st
           b1,b2 // b1,b4 & b1,b2 _|_ b4,b3) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b1,b2 _|_ b3,b4 & b3 <> b4);
end;

:: ANALMETR:dfs 8
definiens
  let a1 be non empty ParOrtStr;
To prove
     a1 is OrtAfSp-like
it is sufficient to prove
  thus AffinStruct(#the carrier of a1,the CONGR of a1#) is non empty non trivial AffinSpace-like AffinStruct &
     (for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
     (b1,b2 _|_ b1,b2 implies b1 = b2) &
      b1,b2 _|_ b3,b3 &
      (b1,b2 _|_ b3,b4 implies b1,b2 _|_ b4,b3 & b3,b4 _|_ b1,b2) &
      (b1,b2 _|_ b5,b6 & b1,b2 // b7,b8 & not b5,b6 _|_ b7,b8 implies b1 = b2) &
      (b1,b2 _|_ b5,b6 & b1,b2 _|_ b5,b8 implies b1,b2 _|_ b6,b8)) &
     (for b1, b2, b3 being Element of the carrier of a1
           st b1 <> b2
        holds ex b4 being Element of the carrier of a1 st
           b1,b2 // b1,b4 & b1,b2 _|_ b4,b3) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b1,b2 _|_ b3,b4 & b3 <> b4);

:: ANALMETR:def 9
theorem
for b1 being non empty ParOrtStr holds
      b1 is OrtAfSp-like
   iff
      AffinStruct(#the carrier of b1,the CONGR of b1#) is non empty non trivial AffinSpace-like AffinStruct &
       (for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1 holds
       (b2,b3 _|_ b2,b3 implies b2 = b3) &
        b2,b3 _|_ b4,b4 &
        (b2,b3 _|_ b4,b5 implies b2,b3 _|_ b5,b4 & b4,b5 _|_ b2,b3) &
        (b2,b3 _|_ b6,b7 & b2,b3 // b8,b9 & not b6,b7 _|_ b8,b9 implies b2 = b3) &
        (b2,b3 _|_ b6,b7 & b2,b3 _|_ b6,b9 implies b2,b3 _|_ b7,b9)) &
       (for b2, b3, b4 being Element of the carrier of b1
             st b2 <> b3
          holds ex b5 being Element of the carrier of b1 st
             b2,b3 // b2,b5 & b2,b3 _|_ b5,b4) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       ex b5 being Element of the carrier of b1 st
          b2,b3 _|_ b4,b5 & b4 <> b5);

:: ANALMETR:exreg 3
registration
  cluster non empty strict OrtAfSp-like ParOrtStr;
end;

:: ANALMETR:modenot 1
definition
  mode OrtAfSp is non empty OrtAfSp-like ParOrtStr;
end;

:: ANALMETR:th 44
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 AMSpace(b1,b2,b3) is non empty OrtAfSp-like ParOrtStr;

:: ANALMETR:attrnot 3 => ANALMETR:attr 3
definition
  let a1 be non empty ParOrtStr;
  attr a1 is OrtAfPl-like means
    AffinStruct(#the carrier of a1,the CONGR of a1#) is non empty non trivial AffinSpace-like 2-dimensional AffinStruct &
     (for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
     (b1,b2 _|_ b1,b2 implies b1 = b2) &
      b1,b2 _|_ b3,b3 &
      (b1,b2 _|_ b3,b4 implies b1,b2 _|_ b4,b3 & b3,b4 _|_ b1,b2) &
      (b1,b2 _|_ b5,b6 & b1,b2 // b7,b8 & not b5,b6 _|_ b7,b8 implies b1 = b2) &
      (b1,b2 _|_ b5,b6 & b1,b2 _|_ b7,b8 & not b5,b6 // b7,b8 implies b1 = b2)) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b1,b2 _|_ b3,b4 & b3 <> b4);
end;

:: ANALMETR:dfs 9
definiens
  let a1 be non empty ParOrtStr;
To prove
     a1 is OrtAfPl-like
it is sufficient to prove
  thus AffinStruct(#the carrier of a1,the CONGR of a1#) is non empty non trivial AffinSpace-like 2-dimensional AffinStruct &
     (for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
     (b1,b2 _|_ b1,b2 implies b1 = b2) &
      b1,b2 _|_ b3,b3 &
      (b1,b2 _|_ b3,b4 implies b1,b2 _|_ b4,b3 & b3,b4 _|_ b1,b2) &
      (b1,b2 _|_ b5,b6 & b1,b2 // b7,b8 & not b5,b6 _|_ b7,b8 implies b1 = b2) &
      (b1,b2 _|_ b5,b6 & b1,b2 _|_ b7,b8 & not b5,b6 // b7,b8 implies b1 = b2)) &
     (for b1, b2, b3 being Element of the carrier of a1 holds
     ex b4 being Element of the carrier of a1 st
        b1,b2 _|_ b3,b4 & b3 <> b4);

:: ANALMETR:def 10
theorem
for b1 being non empty ParOrtStr holds
      b1 is OrtAfPl-like
   iff
      AffinStruct(#the carrier of b1,the CONGR of b1#) is non empty non trivial AffinSpace-like 2-dimensional AffinStruct &
       (for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1 holds
       (b2,b3 _|_ b2,b3 implies b2 = b3) &
        b2,b3 _|_ b4,b4 &
        (b2,b3 _|_ b4,b5 implies b2,b3 _|_ b5,b4 & b4,b5 _|_ b2,b3) &
        (b2,b3 _|_ b6,b7 & b2,b3 // b8,b9 & not b6,b7 _|_ b8,b9 implies b2 = b3) &
        (b2,b3 _|_ b6,b7 & b2,b3 _|_ b8,b9 & not b6,b7 // b8,b9 implies b2 = b3)) &
       (for b2, b3, b4 being Element of the carrier of b1 holds
       ex b5 being Element of the carrier of b1 st
          b2,b3 _|_ b4,b5 & b4 <> b5);

:: ANALMETR:exreg 4
registration
  cluster non empty strict OrtAfPl-like ParOrtStr;
end;

:: ANALMETR:modenot 2
definition
  mode OrtAfPl is non empty OrtAfPl-like ParOrtStr;
end;

:: ANALMETR:th 46
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 AMSpace(b1,b2,b3) is non empty OrtAfPl-like ParOrtStr;

:: ANALMETR:th 47
theorem
for b1 being non empty ParOrtStr
for b2 being set holds
      b2 is Element of the carrier of b1
   iff
      b2 is Element of the carrier of Af b1;

:: ANALMETR:th 48
theorem
for b1 being non empty ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8, b9 being Element of the carrier of Af b1
      st b2 = b6 & b3 = b7 & b4 = b8 & b5 = b9
   holds    b2,b3 // b4,b5
   iff
      b6,b7 // b8,b9;

:: ANALMETR:funcreg 3
registration
  let a1 be non empty OrtAfSp-like ParOrtStr;
  cluster Af a1 -> non trivial strict AffinSpace-like;
end;

:: ANALMETR:funcreg 4
registration
  let a1 be non empty OrtAfPl-like ParOrtStr;
  cluster Af a1 -> non trivial strict AffinSpace-like 2-dimensional;
end;

:: ANALMETR:th 49
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
   b1 is non empty OrtAfSp-like ParOrtStr;

:: ANALMETR:condreg 1
registration
  cluster non empty OrtAfPl-like -> OrtAfSp-like (ParOrtStr);
end;

:: ANALMETR:th 50
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
      st Af b1 is non empty non trivial AffinSpace-like 2-dimensional AffinStruct
   holds b1 is non empty OrtAfPl-like ParOrtStr;

:: ANALMETR:th 51
theorem
for b1 being non empty ParOrtStr holds
      b1 is OrtAfPl-like
   iff
      (ex b2, b3 being Element of the carrier of b1 st
          b2 <> b3) &
       (for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1 holds
       b2,b3 // b3,b2 &
        b2,b3 // b4,b4 &
        (b2,b3 // b6,b7 & b2,b3 // b8,b9 & not b6,b7 // b8,b9 implies b2 = b3) &
        (b2,b3 // b2,b4 implies b3,b2 // b3,b4) &
        (ex b10 being Element of the carrier of b1 st
           b2,b3 // b4,b10 & b2,b4 // b3,b10) &
        (ex b10, b11, b12 being Element of the carrier of b1 st
           not b10,b11 // b10,b12) &
        (ex b10 being Element of the carrier of b1 st
           b2,b3 // b4,b10 & b4 <> b10) &
        (b2,b3 // b3,b5 & b3 <> b2 implies ex b10 being Element of the carrier of b1 st
           b4,b3 // b3,b10 & b4,b2 // b5,b10) &
        (b2,b3 _|_ b2,b3 implies b2 = b3) &
        b2,b3 _|_ b4,b4 &
        (b2,b3 _|_ b4,b5 implies b2,b3 _|_ b5,b4 & b4,b5 _|_ b2,b3) &
        (b2,b3 _|_ b6,b7 & b2,b3 // b8,b9 & not b6,b7 _|_ b8,b9 implies b2 = b3) &
        (b2,b3 _|_ b6,b7 & b2,b3 _|_ b8,b9 & not b6,b7 // b8,b9 implies b2 = b3) &
        (ex b10 being Element of the carrier of b1 st
           b2,b3 _|_ b4,b10 & b4 <> b10) &
        (not b2,b3 // b4,b5 implies ex b10 being Element of the carrier of b1 st
           b2,b3 // b2,b10 & b4,b5 // b4,b10));

:: ANALMETR:prednot 5 => ANALMETR:pred 5
definition
  let a1 be non empty ParOrtStr;
  let a2, a3, a4 be Element of the carrier of a1;
  pred LIN A2,A3,A4 means
    a2,a3 // a2,a4;
end;

:: ANALMETR:dfs 10
definiens
  let a1 be non empty ParOrtStr;
  let a2, a3, a4 be Element of the carrier of a1;
To prove
     LIN a2,a3,a4
it is sufficient to prove
  thus a2,a3 // a2,a4;

:: ANALMETR:def 11
theorem
for b1 being non empty ParOrtStr
for b2, b3, b4 being Element of the carrier of b1 holds
   LIN b2,b3,b4
iff
   b2,b3 // b2,b4;

:: ANALMETR:funcnot 4 => ANALMETR:func 4
definition
  let a1 be non empty ParOrtStr;
  let a2, a3 be Element of the carrier of a1;
  func Line(A2,A3) -> Element of bool the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
          b1 in it
       iff
          LIN a2,a3,b1;
end;

:: ANALMETR:def 12
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
      b4 = Line(b2,b3)
   iff
      for b5 being Element of the carrier of b1 holds
            b5 in b4
         iff
            LIN b2,b3,b5;

:: ANALMETR:attrnot 4 => ANALMETR:attr 4
definition
  let a1 be non empty ParOrtStr;
  let a2 be Element of bool the carrier of a1;
  attr a2 is being_line means
    ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a2 = Line(b1,b2);
end;

:: ANALMETR:dfs 12
definiens
  let a1 be non empty ParOrtStr;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is being_line
it is sufficient to prove
  thus ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a2 = Line(b1,b2);

:: ANALMETR:def 13
theorem
for b1 being non empty ParOrtStr
for b2 being Element of bool the carrier of b1 holds
      b2 is being_line(b1)
   iff
      ex b3, b4 being Element of the carrier of b1 st
         b3 <> b4 & b2 = Line(b3,b4);

:: ANALMETR:prednot 6 => ANALMETR:attr 4
notation
  let a1 be non empty ParOrtStr;
  let a2 be Element of bool the carrier of a1;
  synonym a2 is_line for being_line;
end;

:: ANALMETR:th 55
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4 being Element of the carrier of b1
for b5, b6, b7 being Element of the carrier of Af b1
      st b2 = b5 & b3 = b6 & b4 = b7
   holds    LIN b2,b3,b4
   iff
      LIN b5,b6,b7;

:: ANALMETR:th 56
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of the carrier of Af b1
      st b2 = b4 & b3 = b5
   holds Line(b2,b3) = Line(b4,b5);

:: ANALMETR:th 57
theorem
for b1 being non empty ParOrtStr
for b2 being set holds
      b2 is Element of bool the carrier of b1
   iff
      b2 is Element of bool the carrier of Af b1;

:: ANALMETR:th 58
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of Af b1
      st b2 = b3
   holds    b2 is being_line(b1)
   iff
      b3 is being_line(Af b1);

:: ANALMETR:prednot 7 => ANALMETR:pred 6
definition
  let a1 be non empty ParOrtStr;
  let a2, a3 be Element of the carrier of a1;
  let a4 be Element of bool the carrier of a1;
  pred A2,A3 _|_ A4 means
    ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a4 = Line(b1,b2) & a2,a3 _|_ b1,b2;
end;

:: ANALMETR:dfs 13
definiens
  let a1 be non empty ParOrtStr;
  let a2, a3 be Element of the carrier of a1;
  let a4 be Element of bool the carrier of a1;
To prove
     a2,a3 _|_ a4
it is sufficient to prove
  thus ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a4 = Line(b1,b2) & a2,a3 _|_ b1,b2;

:: ANALMETR:def 14
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
      b2,b3 _|_ b4
   iff
      ex b5, b6 being Element of the carrier of b1 st
         b5 <> b6 & b4 = Line(b5,b6) & b2,b3 _|_ b5,b6;

:: ANALMETR:prednot 8 => ANALMETR:pred 7
definition
  let a1 be non empty ParOrtStr;
  let a2, a3 be Element of bool the carrier of a1;
  pred A2 _|_ A3 means
    ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a2 = Line(b1,b2) & b1,b2 _|_ a3;
end;

:: ANALMETR:dfs 14
definiens
  let a1 be non empty ParOrtStr;
  let a2, a3 be Element of bool the carrier of a1;
To prove
     a2 _|_ a3
it is sufficient to prove
  thus ex b1, b2 being Element of the carrier of a1 st
       b1 <> b2 & a2 = Line(b1,b2) & b1,b2 _|_ a3;

:: ANALMETR:def 15
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of bool the carrier of b1 holds
   b2 _|_ b3
iff
   ex b4, b5 being Element of the carrier of b1 st
      b4 <> b5 & b2 = Line(b4,b5) & b4,b5 _|_ b3;

:: ANALMETR:prednot 9 => ANALMETR:pred 8
definition
  let a1 be non empty ParOrtStr;
  let a2, a3 be Element of bool the carrier of a1;
  pred A2 // A3 means
    ex b1, b2, b3, b4 being Element of the carrier of a1 st
       b1 <> b2 & b3 <> b4 & a2 = Line(b1,b2) & a3 = Line(b3,b4) & b1,b2 // b3,b4;
end;

:: ANALMETR:dfs 15
definiens
  let a1 be non empty ParOrtStr;
  let a2, a3 be Element of bool the carrier of a1;
To prove
     a2 // a3
it is sufficient to prove
  thus ex b1, b2, b3, b4 being Element of the carrier of a1 st
       b1 <> b2 & b3 <> b4 & a2 = Line(b1,b2) & a3 = Line(b3,b4) & b1,b2 // b3,b4;

:: ANALMETR:def 16
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of bool the carrier of b1 holds
   b2 // b3
iff
   ex b4, b5, b6, b7 being Element of the carrier of b1 st
      b4 <> b5 & b6 <> b7 & b2 = Line(b4,b5) & b3 = Line(b6,b7) & b4,b5 // b6,b7;

:: ANALMETR:th 62
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1 holds
(b2,b3 _|_ b4 implies b4 is being_line(b1)) & (b4 _|_ b5 implies b4 is being_line(b1) & b5 is being_line(b1));

:: ANALMETR:th 63
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of bool the carrier of b1 holds
   b2 _|_ b3
iff
   ex b4, b5, b6, b7 being Element of the carrier of b1 st
      b4 <> b5 & b6 <> b7 & b2 = Line(b4,b5) & b3 = Line(b6,b7) & b4,b5 _|_ b6,b7;

:: ANALMETR:th 64
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of bool the carrier of Af b1
      st b2 = b4 & b3 = b5
   holds    b2 // b3
   iff
      b4 // b5;

:: ANALMETR:th 65
theorem
for b1 being non empty OrtAfSp-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 b3,b3 _|_ b2;

:: ANALMETR:th 66
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4, b5, b6 being Element of the carrier of b1
      st b3,b4 _|_ b2 & (b3,b4 // b5,b6 or b5,b6 // b3,b4) & b3 <> b4
   holds b5,b6 _|_ b2;

:: ANALMETR:th 67
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
      st b3,b4 _|_ b2
   holds b4,b3 _|_ b2;

:: ANALMETR:prednot 10 => ANALMETR:pred 9
definition
  let a1 be non empty OrtAfSp-like ParOrtStr;
  let a2, a3 be Element of bool the carrier of a1;
  redefine pred a2 // a3;
  symmetry;
::  for a1 being non empty OrtAfSp-like ParOrtStr
::  for a2, a3 being Element of bool the carrier of a1
::        st a2 // a3
::     holds a3 // a2;
end;

:: ANALMETR:th 69
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of the carrier of b1
      st b4,b5 _|_ b2 & b2 // b3
   holds b4,b5 _|_ b3;

:: ANALMETR:th 71
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
      st b3 in b2 & b4 in b2 & b3,b4 _|_ b2
   holds b3 = b4;

:: ANALMETR:prednot 11 => ANALMETR:pred 10
definition
  let a1 be non empty OrtAfSp-like ParOrtStr;
  let a2, a3 be Element of bool the carrier of a1;
  redefine pred a2 _|_ a3;
  symmetry;
::  for a1 being non empty OrtAfSp-like ParOrtStr
::  for a2, a3 being Element of bool the carrier of a1
::        st a2 _|_ a3
::     holds a3 _|_ a2;
  irreflexivity;
::  for a1 being non empty OrtAfSp-like ParOrtStr
::  for a2 being Element of bool the carrier of a1 holds
::     not a2 _|_ a2;
end;

:: ANALMETR:th 73
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4 being Element of bool the carrier of b1
      st b2 _|_ b3 & b2 // b4
   holds b4 _|_ b3;

:: ANALMETR:th 75
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4, b5, b6 being Element of the carrier of b1
      st b3 in b2 & b4 in b2 & b5,b6 _|_ b2
   holds b5,b6 _|_ b3,b4 & b3,b4 _|_ b5,b6;

:: ANALMETR:th 76
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
      st b3 in b2 & b4 in b2 & b3 <> b4 & b2 is being_line(b1)
   holds b2 = Line(b3,b4);

:: ANALMETR:th 77
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4, b5, b6 being Element of the carrier of b1
      st b3 in b2 & b4 in b2 & b3 <> b4 & b2 is being_line(b1) & (b3,b4 _|_ b5,b6 or b5,b6 _|_ b3,b4)
   holds b5,b6 _|_ b2;

:: ANALMETR:th 78
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3 being Element of bool the carrier of b1
for b4, b5, b6, b7 being Element of the carrier of b1
      st b4 in b2 & b5 in b2 & b6 in b3 & b7 in b3 & b2 _|_ b3
   holds b4,b5 _|_ b6,b7;

:: ANALMETR:th 79
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4, b5 being Element of bool the carrier of b1
for b6, b7, b8 being Element of the carrier of b1
      st b6 in b2 & b6 in b3 & b7 in b2 & b8 in b3 & b7 <> b8 & b7 in b4 & b8 in b4 & b5 _|_ b2 & b5 _|_ b3 & b4 is being_line(b1)
   holds b5 _|_ b4;

:: ANALMETR:th 80
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3 _|_ b4,b4 & b4,b4 _|_ b2,b3 & b2,b3 // b4,b4 & b4,b4 // b2,b3;

:: ANALMETR:th 81
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 // b4,b5
   holds b2,b3 // b5,b4 & b3,b2 // b4,b5 & b3,b2 // b5,b4 & b4,b5 // b2,b3 & b4,b5 // b3,b2 & b5,b4 // b2,b3 & b5,b4 // b3,b2;

:: ANALMETR:th 82
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2 <> b3 &
         ((b2,b3 // b4,b5 implies not b2,b3 // b6,b7) & (b2,b3 // b4,b5 implies not b6,b7 // b2,b3) & (b4,b5 // b2,b3 implies not b6,b7 // b2,b3) implies b4,b5 // b2,b3 & b2,b3 // b6,b7)
   holds b4,b5 // b6,b7;

:: ANALMETR:th 83
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 _|_ b4,b5
   holds b2,b3 _|_ b5,b4 & b3,b2 _|_ b4,b5 & b3,b2 _|_ b5,b4 & b4,b5 _|_ b2,b3 & b4,b5 _|_ b3,b2 & b5,b4 _|_ b2,b3 & b5,b4 _|_ b3,b2;

:: ANALMETR:th 84
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2 <> b3 &
         ((b2,b3 // b4,b5 implies not b2,b3 _|_ b6,b7) & (b2,b3 // b6,b7 implies not b2,b3 _|_ b4,b5) & (b2,b3 // b4,b5 implies not b6,b7 _|_ b2,b3) & (b2,b3 // b6,b7 implies not b4,b5 _|_ b2,b3) & (b4,b5 // b2,b3 implies not b6,b7 _|_ b2,b3) & (b6,b7 // b2,b3 implies not b4,b5 _|_ b2,b3) & (b4,b5 // b2,b3 implies not b2,b3 _|_ b6,b7) implies b6,b7 // b2,b3 & b2,b3 _|_ b4,b5)
   holds b4,b5 _|_ b6,b7;

:: ANALMETR:th 85
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2 <> b3 &
         ((b2,b3 _|_ b4,b5 implies not b2,b3 _|_ b6,b7) & (b2,b3 _|_ b4,b5 implies not b6,b7 _|_ b2,b3) & (b4,b5 _|_ b2,b3 implies not b6,b7 _|_ b2,b3) implies b4,b5 _|_ b2,b3 & b2,b3 _|_ b6,b7)
   holds b4,b5 // b6,b7;

:: ANALMETR:th 86
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3 being Element of bool the carrier of b1
for b4, b5, b6, b7 being Element of the carrier of b1
      st b4 in b2 & b5 in b2 & b4 <> b5 & b2 is being_line(b1) & b6 in b3 & b7 in b3 & b6 <> b7 & b3 is being_line(b1) & b4,b5 // b6,b7
   holds b2 // b3;

:: ANALMETR:th 87
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4 being Element of bool the carrier of b1
      st b2 _|_ b3 & b4 _|_ b3
   holds b2 // b4;

:: ANALMETR:th 88
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3 being Element of bool the carrier of b1
      st b2 _|_ b3
   holds ex b4 being Element of the carrier of b1 st
      b4 in b2 & b4 in b3;

:: ANALMETR:th 89
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 _|_ b4,b5
   holds ex b6 being Element of the carrier of b1 st
      LIN b2,b3,b6 & LIN b4,b5,b6;

:: ANALMETR:th 90
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
      st b3,b4 _|_ b2
   holds ex b5 being Element of the carrier of b1 st
      LIN b3,b4,b5 & b5 in b2;

:: ANALMETR:th 91
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
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 & LIN b3,b4,b5;

:: ANALMETR:th 92
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 the carrier of b1 st
      b3,b4 _|_ b2 & b4 in b2;