Article RLVECT_4, MML version 4.99.1005

:: RLVECT_4:sch 1
scheme RLVECT_4:sch 1
{F1 -> non empty set,
  F2 -> non empty set,
  F3 -> Element of F1(),
  F4 -> Element of F1(),
  F5 -> Element of F1(),
  F6 -> Element of F2(),
  F7 -> Element of F2(),
  F8 -> Element of F2(),
  F9 -> Element of F2()}:
ex b1 being Function-like quasi_total Relation of F1(),F2() st
   b1 . F3() = F6() &
    b1 . F4() = F7() &
    b1 . F5() = F8() &
    (for b2 being Element of F1()
          st b2 <> F3() & b2 <> F4() & b2 <> F5()
       holds b1 . b2 = F9(b2))
provided
   F3() <> F4()
and
   F3() <> F5()
and
   F4() <> F5();


:: RLVECT_4:sch 2
scheme RLVECT_4:sch 2
{F1 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct,
  F2 -> Element of the carrier of F1(),
  F3 -> Element of REAL}:
ex b1 being Linear_Combination of {F2()} st
   b1 . F2() = F3()


:: RLVECT_4:sch 3
scheme RLVECT_4:sch 3
{F1 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct,
  F2 -> Element of the carrier of F1(),
  F3 -> Element of the carrier of F1(),
  F4 -> Element of REAL,
  F5 -> Element of REAL}:
ex b1 being Linear_Combination of {F2(),F3()} st
   b1 . F2() = F4() & b1 . F3() = F5()
provided
   F2() <> F3();


:: RLVECT_4:sch 4
scheme RLVECT_4:sch 4
{F1 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct,
  F2 -> Element of the carrier of F1(),
  F3 -> Element of the carrier of F1(),
  F4 -> Element of the carrier of F1(),
  F5 -> Element of REAL,
  F6 -> Element of REAL,
  F7 -> Element of REAL}:
ex b1 being Linear_Combination of {F2(),F3(),F4()} st
   b1 . F2() = F5() & b1 . F3() = F6() & b1 . F4() = F7()
provided
   F2() <> F3()
and
   F2() <> F4()
and
   F3() <> F4();


:: RLVECT_4:th 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
(b2 + b3) - b2 = b3 & (b3 + b2) - b2 = b3 & (b2 - b2) + b3 = b3 & (b3 - b2) + b2 = b3 & b2 + (b3 - b2) = b3 & b3 + (b2 - b2) = b3 & b2 - (b2 - b3) = b3;

:: RLVECT_4: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 b2 - b3 = b4 - b3
   holds b2 = b4;

:: RLVECT_4:th 6
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of the carrier of b2 holds
   - (b1 * b3) = (- b1) * b3;

:: RLVECT_4:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1
for b3, b4 being Subspace of b1
      st b3 is Subspace of b4
   holds b2 + b3 c= b2 + b4;

:: RLVECT_4:th 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
for b4 being Subspace of b1
      st b2 in b3 + b4
   holds b3 + b4 = b2 + b4;

:: RLVECT_4: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
for b5 being Linear_Combination of {b2,b3,b4}
      st b2 <> b3 & b2 <> b4 & b3 <> b4
   holds Sum b5 = (((b5 . b2) * b2) + ((b5 . b3) * b3)) + ((b5 . b4) * b4);

:: RLVECT_4:th 10
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
   b2 <> b3 & b2 <> b4 & b3 <> b4 & {b2,b3,b4} is linearly-independent(b1)
iff
   for b5, b6, b7 being Element of REAL
         st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
      holds b5 = 0 & b6 = 0 & b7 = 0;

:: RLVECT_4:th 11
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of the carrier of b2 holds
      b1 in Lin {b3}
   iff
      ex b4 being Element of REAL st
         b1 = b4 * b3;

:: RLVECT_4:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
   b2 in Lin {b2};

:: RLVECT_4:th 13
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of the carrier of b2 holds
   b1 in b3 + Lin {b4}
iff
   ex b5 being Element of REAL st
      b1 = b3 + (b5 * b4);

:: RLVECT_4:th 14
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of the carrier of b2 holds
   b1 in Lin {b3,b4}
iff
   ex b5, b6 being Element of REAL st
      b1 = (b5 * b3) + (b6 * b4);

:: RLVECT_4:th 15
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 in Lin {b2,b3} & b3 in Lin {b2,b3};

:: RLVECT_4:th 16
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4, b5 being Element of the carrier of b2 holds
   b1 in b3 + Lin {b4,b5}
iff
   ex b6, b7 being Element of REAL st
      b1 = (b3 + (b6 * b4)) + (b7 * b5);

:: RLVECT_4:th 17
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4, b5 being Element of the carrier of b2 holds
   b1 in Lin {b3,b4,b5}
iff
   ex b6, b7, b8 being Element of REAL st
      b1 = ((b6 * b3) + (b7 * b4)) + (b8 * b5);

:: RLVECT_4:th 18
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
b2 in Lin {b2,b3,b4} & b3 in Lin {b2,b3,b4} & b4 in Lin {b2,b3,b4};

:: RLVECT_4:th 19
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4, b5, b6 being Element of the carrier of b2 holds
   b1 in b3 + Lin {b4,b5,b6}
iff
   ex b7, b8, b9 being Element of REAL st
      b1 = ((b3 + (b7 * b4)) + (b8 * b5)) + (b9 * b6);

:: RLVECT_4:th 20
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 {b2,b3} is linearly-independent(b1) & b2 <> b3
   holds {b2,b3 - b2} is linearly-independent(b1);

:: RLVECT_4:th 21
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 {b2,b3} is linearly-independent(b1) & b2 <> b3
   holds {b2,b3 + b2} is linearly-independent(b1);

:: RLVECT_4:th 22
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of the carrier of b2
      st {b3,b4} is linearly-independent(b2) & b3 <> b4 & b1 <> 0
   holds {b3,b1 * b4} is linearly-independent(b2);

:: RLVECT_4:th 23
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 {b2,b3} is linearly-independent(b1) & b2 <> b3
   holds {b2,- b3} is linearly-independent(b1);

:: RLVECT_4:th 24
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b4 being Element of the carrier of b3
      st b1 <> b2
   holds {b1 * b4,b2 * b4} is linearly-dependent(b3);

:: RLVECT_4:th 25
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of the carrier of b2
      st b1 <> 1
   holds {b3,b1 * b3} is linearly-dependent(b2);

:: RLVECT_4:th 26
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 {b2,b3,b4} is linearly-independent(b1) & b2 <> b4 & b2 <> b3 & b4 <> b3
   holds {b2,b3,b4 - b2} is linearly-independent(b1);

:: RLVECT_4:th 27
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 {b2,b3,b4} is linearly-independent(b1) & b2 <> b4 & b2 <> b3 & b4 <> b3
   holds {b2,b3 - b2,b4 - b2} is linearly-independent(b1);

:: RLVECT_4:th 28
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 {b2,b3,b4} is linearly-independent(b1) & b2 <> b4 & b2 <> b3 & b4 <> b3
   holds {b2,b3,b4 + b2} is linearly-independent(b1);

:: RLVECT_4:th 29
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 {b2,b3,b4} is linearly-independent(b1) & b2 <> b4 & b2 <> b3 & b4 <> b3
   holds {b2,b3 + b2,b4 + b2} is linearly-independent(b1);

:: RLVECT_4:th 30
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4, b5 being Element of the carrier of b2
      st {b3,b4,b5} is linearly-independent(b2) & b3 <> b5 & b3 <> b4 & b5 <> b4 & b1 <> 0
   holds {b3,b4,b1 * b5} is linearly-independent(b2);

:: RLVECT_4:th 31
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b4, b5, b6 being Element of the carrier of b3
      st {b4,b5,b6} is linearly-independent(b3) & b4 <> b6 & b4 <> b5 & b6 <> b5 & b1 <> 0 & b2 <> 0
   holds {b4,b1 * b5,b2 * b6} is linearly-independent(b3);

:: RLVECT_4:th 32
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 {b2,b3,b4} is linearly-independent(b1) & b2 <> b4 & b2 <> b3 & b4 <> b3
   holds {b2,b3,- b4} is linearly-independent(b1);

:: RLVECT_4:th 33
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 {b2,b3,b4} is linearly-independent(b1) & b2 <> b4 & b2 <> b3 & b4 <> b3
   holds {b2,- b3,- b4} is linearly-independent(b1);

:: RLVECT_4:th 34
theorem
for b1, b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b4, b5 being Element of the carrier of b3
      st b1 <> b2
   holds {b1 * b4,b2 * b4,b5} is linearly-dependent(b3);

:: RLVECT_4:th 35
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of the carrier of b2
      st b1 <> 1
   holds {b3,b1 * b3,b4} is linearly-dependent(b2);

:: RLVECT_4: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
      st b2 in Lin {b3} & b2 <> 0. b1
   holds Lin {b2} = Lin {b3};

:: RLVECT_4:th 37
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 & {b2,b3} is linearly-independent(b1) & b2 in Lin {b4,b5} & b3 in Lin {b4,b5}
   holds Lin {b4,b5} = Lin {b2,b3} &
    {b4,b5} is linearly-independent(b1) &
    b4 <> b5;

:: RLVECT_4: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 b2 <> 0. b1 & {b3,b2} is linearly-dependent(b1)
   holds ex b4 being Element of REAL st
      b3 = b4 * b2;

:: RLVECT_4:th 39
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 b2 <> b3 & {b2,b3} is linearly-independent(b1) & {b4,b2,b3} is linearly-dependent(b1)
   holds ex b5, b6 being Element of REAL st
      b4 = (b5 * b2) + (b6 * b3);