Article ANALORT, MML version 4.99.1005

:: ANALORT:funcnot 1 => ANALORT:func 1
definition
  let a1 be non empty Abelian addLoopStr;
  let a2, a3 be Element of the carrier of a1;
  redefine func a2 + a3 -> Element of the carrier of a1;
  commutativity;
::  for a1 being non empty Abelian addLoopStr
::  for a2, a3 being Element of the carrier of a1 holds
::  a2 + a3 = a3 + a2;
end;

:: ANALORT:funcnot 2 => ANALORT:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  func Ortm(A2,A3,A4) -> Element of the carrier of a1 equals
    ((pr1(a2,a3,a4)) * a2) + ((- pr2(a2,a3,a4)) * a3);
end;

:: ANALORT:def 1
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 holds
Ortm(b2,b3,b4) = ((pr1(b2,b3,b4)) * b2) + ((- pr2(b2,b3,b4)) * b3);

:: ANALORT:th 1
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
   holds Ortm(b2,b3,b4 + b5) = (Ortm(b2,b3,b4)) + Ortm(b2,b3,b5);

:: ANALORT:th 2
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
for b5 being Element of REAL
      st Gen b2,b3
   holds Ortm(b2,b3,b5 * b4) = b5 * Ortm(b2,b3,b4);

:: ANALORT:th 3
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 Ortm(b2,b3,0. b1) = 0. b1;

:: ANALORT:th 4
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
   holds Ortm(b2,b3,- b4) = - Ortm(b2,b3,b4);

:: ANALORT: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 b2,b3
   holds Ortm(b2,b3,b4 - b5) = (Ortm(b2,b3,b4)) - Ortm(b2,b3,b5);

:: ANALORT:th 6
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 & Ortm(b2,b3,b4) = Ortm(b2,b3,b5)
   holds b4 = b5;

:: ANALORT:th 7
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
   holds Ortm(b2,b3,Ortm(b2,b3,b4)) = b4;

:: ANALORT:th 8
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
   holds ex b5 being Element of the carrier of b1 st
      b4 = Ortm(b2,b3,b5);

:: ANALORT:funcnot 3 => ANALORT:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3, a4 be Element of the carrier of a1;
  func Orte(A2,A3,A4) -> Element of the carrier of a1 equals
    ((pr2(a2,a3,a4)) * a2) + ((- pr1(a2,a3,a4)) * a3);
end;

:: ANALORT:def 2
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 holds
Orte(b2,b3,b4) = ((pr2(b2,b3,b4)) * b2) + ((- pr1(b2,b3,b4)) * b3);

:: ANALORT:th 9
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
   holds Orte(b2,b3,- b4) = - Orte(b2,b3,b4);

:: ANALORT: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
      st Gen b2,b3
   holds Orte(b2,b3,b4 + b5) = (Orte(b2,b3,b4)) + Orte(b2,b3,b5);

:: ANALORT: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
      st Gen b2,b3
   holds Orte(b2,b3,b4 - b5) = (Orte(b2,b3,b4)) - Orte(b2,b3,b5);

:: ANALORT:th 12
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
for b5 being Element of REAL
      st Gen b2,b3
   holds Orte(b2,b3,b5 * b4) = b5 * Orte(b2,b3,b4);

:: ANALORT:th 13
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 & Orte(b2,b3,b4) = Orte(b2,b3,b5)
   holds b4 = b5;

:: ANALORT:th 14
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
   holds Orte(b2,b3,Orte(b2,b3,b4)) = - b4;

:: ANALORT: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
   holds ex b5 being Element of the carrier of b1 st
      Orte(b2,b3,b5) = b4;

:: ANALORT:prednot 1 => ANALORT:pred 1
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 A4,A5,A6,A7 are_COrte_wrt A2,A3 means
    Orte(a2,a3,a4),Orte(a2,a3,a5) // a6,a7;
end;

:: ANALORT: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
     a4,a5,a6,a7 are_COrte_wrt a2,a3
it is sufficient to prove
  thus Orte(a2,a3,a4),Orte(a2,a3,a5) // a6,a7;

:: ANALORT: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
   b4,b5,b6,b7 are_COrte_wrt b2,b3
iff
   Orte(b2,b3,b4),Orte(b2,b3,b5) // b6,b7;

:: ANALORT:prednot 2 => ANALORT:pred 2
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 A4,A5,A6,A7 are_COrtm_wrt A2,A3 means
    Ortm(a2,a3,a4),Ortm(a2,a3,a5) // a6,a7;
end;

:: ANALORT:dfs 4
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
     a4,a5,a6,a7 are_COrtm_wrt a2,a3
it is sufficient to prove
  thus Ortm(a2,a3,a4),Ortm(a2,a3,a5) // a6,a7;

:: ANALORT:def 4
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
   b4,b5,b6,b7 are_COrtm_wrt b2,b3
iff
   Ortm(b2,b3,b4),Ortm(b2,b3,b5) // b6,b7;

:: ANALORT:th 16
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
      st Gen b2,b3 & b4,b5 // b6,b7
   holds Orte(b2,b3,b4),Orte(b2,b3,b5) // Orte(b2,b3,b6),Orte(b2,b3,b7);

:: ANALORT:th 17
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
      st Gen b2,b3 & b4,b5 // b6,b7
   holds Ortm(b2,b3,b4),Ortm(b2,b3,b5) // Ortm(b2,b3,b6),Ortm(b2,b3,b7);

:: ANALORT:th 18
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
      st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3
   holds b6,b7,b5,b4 are_COrte_wrt b2,b3;

:: ANALORT:th 19
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
      st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3
   holds b6,b7,b4,b5 are_COrtm_wrt b2,b3;

:: ANALORT:th 20
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 holds
b2,b2,b3,b4 are_COrte_wrt b5,b6;

:: ANALORT:th 21
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 holds
b2,b2,b3,b4 are_COrtm_wrt b5,b6;

:: ANALORT:th 22
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 holds
b2,b3,b4,b4 are_COrte_wrt b5,b6;

:: ANALORT:th 23
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 holds
b2,b3,b4,b4 are_COrtm_wrt b5,b6;

:: ANALORT: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
      st Gen b2,b3
   holds b4,b5,Orte(b2,b3,b4),Orte(b2,b3,b5) are_Ort_wrt b2,b3;

:: ANALORT:th 25
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,Orte(b4,b5,b2),Orte(b4,b5,b3) are_COrte_wrt b4,b5;

:: ANALORT:th 26
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,Ortm(b4,b5,b2),Ortm(b4,b5,b3) are_COrtm_wrt b4,b5;

:: ANALORT:th 27
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
      st Gen b2,b3
   holds    b4,b5 // b6,b7
   iff
      ex b8, b9 being Element of the carrier of b1 st
         b8 <> b9 & b8,b9,b4,b5 are_COrte_wrt b2,b3 & b8,b9,b6,b7 are_COrte_wrt b2,b3;

:: ANALORT:th 28
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
      st Gen b2,b3
   holds    b4,b5 // b6,b7
   iff
      ex b8, b9 being Element of the carrier of b1 st
         b8 <> b9 & b8,b9,b4,b5 are_COrtm_wrt b2,b3 & b8,b9,b6,b7 are_COrtm_wrt b2,b3;

:: ANALORT:th 29
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
      st Gen b2,b3
   holds    b4,b5,b6,b7 are_Ort_wrt b2,b3
   iff
      (b4,b5,b6,b7 are_COrte_wrt b2,b3 or b4,b5,b7,b6 are_COrte_wrt b2,b3);

:: ANALORT:th 30
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
      st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b4,b5,b7,b6 are_COrte_wrt b2,b3 & b4 <> b5
   holds b6 = b7;

:: ANALORT: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
      st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b4,b5,b7,b6 are_COrtm_wrt b2,b3 & b4 <> b5
   holds b6 = b7;

:: ANALORT: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, b8 being Element of the carrier of b1
      st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b4,b5,b6,b8 are_COrte_wrt b2,b3 & not b4,b5,b7,b8 are_COrte_wrt b2,b3
   holds b4,b5,b8,b7 are_COrte_wrt b2,b3;

:: ANALORT:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b4,b5,b6,b8 are_COrtm_wrt b2,b3 & not b4,b5,b7,b8 are_COrtm_wrt b2,b3
   holds b4,b5,b8,b7 are_COrtm_wrt b2,b3;

:: ANALORT:th 34
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
      st b2,b3,b4,b5 are_COrte_wrt b6,b7
   holds b3,b2,b5,b4 are_COrte_wrt b6,b7;

:: ANALORT:th 35
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
      st b2,b3,b4,b5 are_COrtm_wrt b6,b7
   holds b3,b2,b5,b4 are_COrtm_wrt b6,b7;

:: ANALORT:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b4,b5,b7,b8 are_COrte_wrt b2,b3
   holds b4,b5,b6,b8 are_COrte_wrt b2,b3;

:: ANALORT:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b4,b5,b7,b8 are_COrtm_wrt b2,b3
   holds b4,b5,b6,b8 are_COrtm_wrt b2,b3;

:: ANALORT: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
   st Gen b2,b3
for b4, b5, b6 being Element of the carrier of b1 holds
ex b7 being Element of the carrier of b1 st
   b6 <> b7 & b6,b7,b4,b5 are_COrte_wrt b2,b3;

:: ANALORT: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
   st Gen b2,b3
for b4, b5, b6 being Element of the carrier of b1 holds
ex b7 being Element of the carrier of b1 st
   b6 <> b7 & b6,b7,b4,b5 are_COrtm_wrt b2,b3;

:: ANALORT: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
   st Gen b2,b3
for b4, b5, b6 being Element of the carrier of b1 holds
ex b7 being Element of the carrier of b1 st
   b6 <> b7 & b4,b5,b6,b7 are_COrte_wrt b2,b3;

:: ANALORT: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
   st Gen b2,b3
for b4, b5, b6 being Element of the carrier of b1 holds
ex b7 being Element of the carrier of b1 st
   b6 <> b7 & b4,b5,b6,b7 are_COrtm_wrt b2,b3;

:: ANALORT:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
      st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b8,b9,b6,b7 are_COrte_wrt b2,b3 & b8,b9,b10,b11 are_COrte_wrt b2,b3 & b8 <> b9 & b6 <> b7
   holds b4,b5,b10,b11 are_COrte_wrt b2,b3;

:: ANALORT:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
      st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b8,b9,b6,b7 are_COrtm_wrt b2,b3 & b8,b9,b10,b11 are_COrtm_wrt b2,b3 & b8 <> b9 & b6 <> b7
   holds b4,b5,b10,b11 are_COrtm_wrt b2,b3;

:: ANALORT:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
      st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b6,b7,b8,b9 are_COrte_wrt b2,b3 & b10,b11,b8,b9 are_COrte_wrt b2,b3 & not b4,b5,b10,b11 are_COrte_wrt b2,b3 & b6 <> b7
   holds b8 = b9;

:: ANALORT:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
      st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b6,b7,b8,b9 are_COrtm_wrt b2,b3 & b10,b11,b8,b9 are_COrtm_wrt b2,b3 & not b4,b5,b10,b11 are_COrtm_wrt b2,b3 & b6 <> b7
   holds b8 = b9;

:: ANALORT:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
      st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b6,b7,b8,b9 are_COrte_wrt b2,b3 & b4,b5,b10,b11 are_COrte_wrt b2,b3 & not b10,b11,b8,b9 are_COrte_wrt b2,b3 & b6 <> b7
   holds b4 = b5;

:: ANALORT:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
      st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b6,b7,b8,b9 are_COrtm_wrt b2,b3 & b4,b5,b10,b11 are_COrtm_wrt b2,b3 & not b10,b11,b8,b9 are_COrtm_wrt b2,b3 & b6 <> b7
   holds b4 = b5;

:: ANALORT:th 50
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, b7, b8 being Element of the carrier of b1
      st not b4,b7,b5,b6 are_COrte_wrt b2,b3 & not b4,b7,b6,b5 are_COrte_wrt b2,b3 & b6,b8,b6,b5 are_COrte_wrt b2,b3
   holds ex b9 being Element of the carrier of b1 st
      (b4,b7,b4,b9 are_COrte_wrt b2,b3 or b4,b7,b9,b4 are_COrte_wrt b2,b3) & (b6,b8,b6,b9 are_COrte_wrt b2,b3 or b6,b8,b9,b6 are_COrte_wrt b2,b3);

:: ANALORT:th 51
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 ex b4, b5, b6 being Element of the carrier of b1 st
      b4,b5,b4,b6 are_COrte_wrt b2,b3 &
       (for b7, b8 being Element of the carrier of b1
             st b7,b8,b4,b5 are_COrte_wrt b2,b3 & (b7,b8,b4,b6 are_COrte_wrt b2,b3 or b7,b8,b6,b4 are_COrte_wrt b2,b3)
          holds b7 = b8);

:: ANALORT:th 52
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, b7, b8 being Element of the carrier of b1
      st not b4,b7,b5,b6 are_COrtm_wrt b2,b3 & not b4,b7,b6,b5 are_COrtm_wrt b2,b3 & b6,b8,b6,b5 are_COrtm_wrt b2,b3
   holds ex b9 being Element of the carrier of b1 st
      (b4,b7,b4,b9 are_COrtm_wrt b2,b3 or b4,b7,b9,b4 are_COrtm_wrt b2,b3) & (b6,b8,b6,b9 are_COrtm_wrt b2,b3 or b6,b8,b9,b6 are_COrtm_wrt b2,b3);

:: ANALORT:th 53
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 ex b4, b5, b6 being Element of the carrier of b1 st
      b4,b5,b4,b6 are_COrtm_wrt b2,b3 &
       (for b7, b8 being Element of the carrier of b1
             st b7,b8,b4,b5 are_COrtm_wrt b2,b3 & (b7,b8,b4,b6 are_COrtm_wrt b2,b3 or b7,b8,b6,b4 are_COrtm_wrt b2,b3)
          holds b7 = b8);

:: ANALORT:funcnot 4 => ANALORT:func 4
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 CORTE(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_COrte_wrt a2,a3;
end;

:: ANALORT:def 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
for b4 being Relation of [:the carrier of b1,the carrier of b1:],[:the carrier of b1,the carrier of b1:] holds
      b4 = CORTE(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_COrte_wrt b2,b3;

:: ANALORT:funcnot 5 => ANALORT:func 5
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 CORTM(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_COrtm_wrt a2,a3;
end;

:: ANALORT:def 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
for b4 being Relation of [:the carrier of b1,the carrier of b1:],[:the carrier of b1,the carrier of b1:] holds
      b4 = CORTM(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_COrtm_wrt b2,b3;

:: ANALORT:funcnot 6 => ANALORT:func 6
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 CESpace(A1,A2,A3) -> strict AffinStruct equals
    AffinStruct(#the carrier of a1,CORTE(a1,a2,a3)#);
end;

:: ANALORT: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
CESpace(b1,b2,b3) = AffinStruct(#the carrier of b1,CORTE(b1,b2,b3)#);

:: ANALORT: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 CESpace(a1,a2,a3) -> non empty strict;
end;

:: ANALORT:funcnot 7 => ANALORT:func 7
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 CMSpace(A1,A2,A3) -> strict AffinStruct equals
    AffinStruct(#the carrier of a1,CORTM(a1,a2,a3)#);
end;

:: ANALORT:def 8
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
CMSpace(b1,b2,b3) = AffinStruct(#the carrier of b1,CORTM(b1,b2,b3)#);

:: ANALORT:funcreg 2
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 CMSpace(a1,a2,a3) -> non empty strict;
end;

:: ANALORT:th 54
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 set holds
      b4 is Element of the carrier of CESpace(b1,b2,b3)
   iff
      b4 is Element of the carrier of b1;

:: ANALORT:th 55
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 set holds
      b4 is Element of the carrier of CMSpace(b1,b2,b3)
   iff
      b4 is Element of the carrier of b1;

:: ANALORT:th 56
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 CESpace(b1,b6,b7)
      st b2 = b8 & b3 = b9 & b4 = b10 & b5 = b11
   holds    b8,b9 // b10,b11
   iff
      b2,b3,b4,b5 are_COrte_wrt b6,b7;

:: ANALORT:th 57
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 CMSpace(b1,b6,b7)
      st b2 = b8 & b3 = b9 & b4 = b10 & b5 = b11
   holds    b8,b9 // b10,b11
   iff
      b2,b3,b4,b5 are_COrtm_wrt b6,b7;