Article ORTSP_1, MML version 4.99.1005

:: ORTSP_1:attrnot 1 => ORTSP_1:attr 1
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed SymStr over a1;
  attr a2 is OrtSp-like means
    for b1, b2, b3, b4, b5 being Element of the carrier of a2
    for b6 being Element of the carrier of a1 holds
       (b1 <> 0. a2 & b2 <> 0. a2 & b3 <> 0. a2 & b4 <> 0. a2 implies ex b7 being Element of the carrier of a2 st
           not b7 <= b1 & not b7 <= b2 & not b7 <= b3 & not b7 <= b4) &
        (b1 <= b2 implies b6 * b1 <= b2) &
        (b2 <= b1 & b3 <= b1 implies b2 + b3 <= b1) &
        (not b2 <= b1 implies ex b7 being Element of the carrier of a1 st
           b5 - (b7 * b2) <= b1) &
        (b1 <= b2 - b3 & b2 <= b3 - b1 implies b3 <= b1 - b2);
end;

:: ORTSP_1:dfs 1
definiens
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed SymStr over a1;
To prove
     a2 is OrtSp-like
it is sufficient to prove
  thus for b1, b2, b3, b4, b5 being Element of the carrier of a2
    for b6 being Element of the carrier of a1 holds
       (b1 <> 0. a2 & b2 <> 0. a2 & b3 <> 0. a2 & b4 <> 0. a2 implies ex b7 being Element of the carrier of a2 st
           not b7 <= b1 & not b7 <= b2 & not b7 <= b3 & not b7 <= b4) &
        (b1 <= b2 implies b6 * b1 <= b2) &
        (b2 <= b1 & b3 <= b1 implies b2 + b3 <= b1) &
        (not b2 <= b1 implies ex b7 being Element of the carrier of a1 st
           b5 - (b7 * b2) <= b1) &
        (b1 <= b2 - b3 & b2 <= b3 - b1 implies b3 <= b1 - b2);

:: ORTSP_1:def 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed SymStr over b1 holds
      b2 is OrtSp-like(b1)
   iff
      for b3, b4, b5, b6, b7 being Element of the carrier of b2
      for b8 being Element of the carrier of b1 holds
         (b3 <> 0. b2 & b4 <> 0. b2 & b5 <> 0. b2 & b6 <> 0. b2 implies ex b9 being Element of the carrier of b2 st
             not b9 <= b3 & not b9 <= b4 & not b9 <= b5 & not b9 <= b6) &
          (b3 <= b4 implies b8 * b3 <= b4) &
          (b4 <= b3 & b5 <= b3 implies b4 + b5 <= b3) &
          (not b4 <= b3 implies ex b9 being Element of the carrier of b1 st
             b7 - (b9 * b4) <= b3) &
          (b3 <= b4 - b5 & b4 <= b5 - b3 implies b5 <= b3 - b4);

:: ORTSP_1:exreg 1
registration
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  cluster non empty right_complementable Abelian add-associative right_zeroed VectSp-like strict OrtSp-like SymStr over a1;
end;

:: ORTSP_1:modenot 1
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  mode OrtSp of a1 is non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over a1;
end;

:: ORTSP_1:th 11
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3 being Element of the carrier of b2 holds
   0. b2 <= b3;

:: ORTSP_1:th 12
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4 being Element of the carrier of b2
      st b3 <= b4
   holds b4 <= b3;

:: ORTSP_1:th 13
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
      st not b3 <= b4 & b5 + b3 <= b4
   holds not b5 <= b4;

:: ORTSP_1:th 14
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
      st not b3 <= b4 & b5 <= b4
   holds not b3 + b5 <= b4;

:: ORTSP_1:th 15
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Element of the carrier of b1
      st not b3 <= b4 & b5 <> 0. b1
   holds not b5 * b3 <= b4 & not b3 <= b5 * b4;

:: ORTSP_1:th 16
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4 being Element of the carrier of b2
      st b3 <= b4
   holds - b3 <= b4;

:: ORTSP_1:th 19
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
      st b3 - b4 <= b5 & b3 - b6 <= b5
   holds b4 - b6 <= b5;

:: ORTSP_1:th 20
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
for b6, b7 being Element of the carrier of b1
      st not b3 <= b4 & b5 - (b6 * b3) <= b4 & b5 - (b7 * b3) <= b4
   holds b6 = b7;

:: ORTSP_1:th 21
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4 being Element of the carrier of b2
      st b3 <= b3 & b4 <= b4
   holds b3 + b4 <= b3 - b4;

:: ORTSP_1:th 22
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
      st (1_ b1) + 1_ b1 <> 0. b1 &
         (ex b3 being Element of the carrier of b2 st
            b3 <> 0. b2)
   holds ex b3 being Element of the carrier of b2 st
      not b3 <= b3;

:: ORTSP_1:funcnot 1 => ORTSP_1:func 1
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over a1;
  let a3, a4, a5 be Element of the carrier of a2;
  assume not a4 <= a3;
  func ProJ(A3,A4,A5) -> Element of the carrier of a1 means
    for b1 being Element of the carrier of a1
          st a5 - (b1 * a4) <= a3
       holds it = b1;
end;

:: ORTSP_1:def 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
   st not b4 <= b3
for b6 being Element of the carrier of b1 holds
      b6 = ProJ(b3,b4,b5)
   iff
      for b7 being Element of the carrier of b1
            st b5 - (b7 * b4) <= b3
         holds b6 = b7;

:: ORTSP_1:th 24
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
      st not b3 <= b4
   holds b5 - ((ProJ(b4,b3,b5)) * b3) <= b4;

:: ORTSP_1:th 25
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
for b6 being Element of the carrier of b1
      st not b3 <= b4
   holds ProJ(b4,b3,b6 * b5) = b6 * ProJ(b4,b3,b5);

:: ORTSP_1:th 26
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
      st not b3 <= b4
   holds ProJ(b4,b3,b5 + b6) = (ProJ(b4,b3,b5)) + ProJ(b4,b3,b6);

:: ORTSP_1:th 27
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
for b6 being Element of the carrier of b1
      st not b3 <= b4 & b6 <> 0. b1
   holds ProJ(b4,b6 * b3,b5) = b6 " * ProJ(b4,b3,b5);

:: ORTSP_1:th 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
for b6 being Element of the carrier of b1
      st not b3 <= b4 & b6 <> 0. b1
   holds ProJ(b6 * b4,b3,b5) = ProJ(b4,b3,b5);

:: ORTSP_1:th 29
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
      st not b3 <= b4 & b5 <= b4
   holds ProJ(b4,b3 + b5,b6) = ProJ(b4,b3,b6) & ProJ(b4,b3,b6 + b5) = ProJ(b4,b3,b6);

:: ORTSP_1:th 30
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
      st not b3 <= b4 & b5 <= b3 & b5 <= b6
   holds ProJ(b4 + b5,b3,b6) = ProJ(b4,b3,b6);

:: ORTSP_1:th 31
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
      st not b3 <= b4 & b5 - b3 <= b4
   holds ProJ(b4,b3,b5) = 1_ b1;

:: ORTSP_1:th 32
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4 being Element of the carrier of b2
      st not b3 <= b4
   holds ProJ(b4,b3,b3) = 1_ b1;

:: ORTSP_1:th 33
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
      st not b3 <= b4
   holds    b5 <= b4
   iff
      ProJ(b4,b3,b5) = 0. b1;

:: ORTSP_1:th 34
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
      st not b3 <= b4 & not b5 <= b4
   holds (ProJ(b4,b3,b6)) * ((ProJ(b4,b3,b5)) ") = ProJ(b4,b5,b6);

:: ORTSP_1:th 35
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
      st not b3 <= b4 & not b5 <= b4
   holds ProJ(b4,b3,b5) = (ProJ(b4,b5,b3)) ";

:: ORTSP_1:th 36
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
      st not b3 <= b4 & b3 <= b5 + b4
   holds ProJ(b4,b3,b5) = - ProJ(b5,b3,b4);

:: ORTSP_1:th 37
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5 being Element of the carrier of b2
      st not b3 <= b4 & not b5 <= b4
   holds ProJ(b5,b4,b3) = (ProJ(b4,b3,b5)) " * ProJ(b3,b4,b5);

:: ORTSP_1:th 38
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
      st not b3 <= b4 & not b3 <= b5 & not b6 <= b4 & not b6 <= b5
   holds (ProJ(b4,b6,b3)) * ProJ(b3,b4,b5) = (ProJ(b6,b4,b5)) * ProJ(b5,b6,b3);

:: ORTSP_1:th 39
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6, b7, b8 being Element of the carrier of b2
      st not b3 <= b4 & not b3 <= b5 & not b6 <= b4 & not b6 <= b5 & not b7 <= b4
   holds ((ProJ(b4,b7,b3)) * ProJ(b3,b4,b5)) * ProJ(b5,b3,b8) = ((ProJ(b4,b7,b6)) * ProJ(b6,b4,b5)) * ProJ(b5,b6,b8);

:: ORTSP_1:th 40
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
      st not b3 <= b4 & not b5 <= b4 & not b6 <= b4
   holds (ProJ(b4,b3,b5)) * ProJ(b5,b4,b6) = (ProJ(b4,b3,b6)) * ProJ(b6,b4,b5);

:: ORTSP_1:funcnot 2 => ORTSP_1:func 2
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over a1;
  let a3, a4, a5, a6 be Element of the carrier of a2;
  assume not a6 <= a5;
  func PProJ(A5,A6,A3,A4) -> Element of the carrier of a1 means
    for b1 being Element of the carrier of a2
          st not b1 <= a5 & not b1 <= a3
       holds it = ((ProJ(a5,a6,b1)) * ProJ(b1,a5,a3)) * ProJ(a3,b1,a4)
    if ex b1 being Element of the carrier of a2 st
       not b1 <= a5 & not b1 <= a3
    otherwise   case for b1 being Element of the carrier of a2
          st not b1 <= a5
       holds b1 <= a3;
    thus it = 0. a1;
  end;
;
end;

:: ORTSP_1:def 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
   st not b6 <= b5
for b7 being Element of the carrier of b1 holds
   (for b8 being Element of the carrier of b2
          st not b8 <= b5
       holds b8 <= b3 or    (b7 = PProJ(b5,b6,b3,b4)
    iff
       for b8 being Element of the carrier of b2
             st not b8 <= b5 & not b8 <= b3
          holds b7 = ((ProJ(b5,b6,b8)) * ProJ(b8,b5,b3)) * ProJ(b3,b8,b4))) &
    (for b8 being Element of the carrier of b2
          st not b8 <= b5
       holds b8 <= b3 implies    (b7 = PProJ(b5,b6,b3,b4)
    iff
       b7 = 0. b1));

:: ORTSP_1:th 43
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
      st not b3 <= b4 & b5 = 0. b2
   holds PProJ(b4,b3,b5,b6) = 0. b1;

:: ORTSP_1:th 44
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
      st not b3 <= b4
   holds    PProJ(b4,b3,b5,b6) = 0. b1
   iff
      b6 <= b5;

:: ORTSP_1:th 45
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
      st not b3 <= b4
   holds PProJ(b4,b3,b5,b6) = PProJ(b4,b3,b6,b5);

:: ORTSP_1:th 46
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6 being Element of the carrier of b2
for b7 being Element of the carrier of b1
      st not b3 <= b4
   holds PProJ(b4,b3,b5,b7 * b6) = b7 * PProJ(b4,b3,b5,b6);

:: ORTSP_1:th 47
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like OrtSp-like SymStr over b1
for b3, b4, b5, b6, b7 being Element of the carrier of b2
      st not b3 <= b4
   holds PProJ(b4,b3,b5,b6 + b7) = (PProJ(b4,b3,b5,b6)) + PProJ(b4,b3,b5,b7);