Article RUSUB_3, MML version 4.99.1005

:: RUSUB_3:funcnot 1 => RUSUB_3:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of bool the carrier of a1;
  func Lin A2 -> strict Subspace of a1 means
    the carrier of it = {Sum b1 where b1 is Linear_Combination of a2: TRUE};
end;

:: RUSUB_3:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being strict Subspace of b1 holds
      b3 = Lin b2
   iff
      the carrier of b3 = {Sum b4 where b4 is Linear_Combination of b2: TRUE};

:: RUSUB_3:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being set holds
      b3 in Lin b2
   iff
      ex b4 being Linear_Combination of b2 st
         b3 = Sum b4;

:: RUSUB_3:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being set
      st b3 in b2
   holds b3 in Lin b2;

:: RUSUB_3:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR holds
   Lin {} the carrier of b1 = (0). b1;

:: RUSUB_3:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
      st Lin b2 = (0). b1 & b2 <> {}
   holds b2 = {0. b1};

:: RUSUB_3:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being strict Subspace of b1
for b3 being Element of bool the carrier of b1
      st b3 = the carrier of b2
   holds Lin b3 = b2;

:: RUSUB_3:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
      st b2 = the carrier of b1
   holds Lin b2 = b1;

:: RUSUB_3:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of bool the carrier of b1
      st b2 c= b3
   holds Lin b2 is Subspace of Lin b3;

:: RUSUB_3:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like UNITSTR
for b2, b3 being Element of bool the carrier of b1
      st Lin b2 = b1 & b2 c= b3
   holds Lin b3 = b1;

:: RUSUB_3:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of bool the carrier of b1 holds
Lin (b2 \/ b3) = (Lin b2) + Lin b3;

:: RUSUB_3:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of bool the carrier of b1 holds
Lin (b2 /\ b3) is Subspace of (Lin b2) /\ Lin b3;

:: RUSUB_3:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
      st b2 is linearly-independent(b1)
   holds ex b3 being Element of bool the carrier of b1 st
      b2 c= b3 &
       b3 is linearly-independent(b1) &
       Lin b3 = UNITSTR(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1,the scalar of b1#);

:: RUSUB_3:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
      st Lin b2 = b1
   holds ex b3 being Element of bool the carrier of b1 st
      b3 c= b2 & b3 is linearly-independent(b1) & Lin b3 = b1;

:: RUSUB_3:modenot 1 => RUSUB_3:mode 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  mode Basis of A1 -> Element of bool the carrier of a1 means
    it is linearly-independent(a1) &
     Lin it = UNITSTR(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1,the scalar of a1#);
end;

:: RUSUB_3:dfs 2
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is Basis of a1
it is sufficient to prove
  thus a2 is linearly-independent(a1) &
     Lin a2 = UNITSTR(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1,the scalar of a1#);

:: RUSUB_3:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1 holds
      b2 is Basis of b1
   iff
      b2 is linearly-independent(b1) &
       Lin b2 = UNITSTR(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1,the scalar of b1#);

:: RUSUB_3:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
      st b2 is linearly-independent(b1)
   holds ex b3 being Basis of b1 st
      b2 c= b3;

:: RUSUB_3:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
      st Lin b2 = b1
   holds ex b3 being Basis of b1 st
      b3 c= b2;

:: RUSUB_3:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
      st b2 is linearly-independent(b1)
   holds ex b3 being Basis of b1 st
      b2 c= b3;

:: RUSUB_3:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Linear_Combination of b1
for b3 being Element of bool the carrier of b1
for b4 being FinSequence of the carrier of b1
      st proj2 b4 c= the carrier of Lin b3
   holds ex b5 being Linear_Combination of b3 st
      Sum (b2 (#) b4) = Sum b5;

:: RUSUB_3:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Linear_Combination of b1
for b3 being Element of bool the carrier of b1
      st Carrier b2 c= the carrier of Lin b3
   holds ex b4 being Linear_Combination of b3 st
      Sum b2 = Sum b4;

:: RUSUB_3:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Linear_Combination of b1
   st Carrier b3 c= the carrier of b2
for b4 being Linear_Combination of b2
      st b4 = b3 | the carrier of b2
   holds Carrier b3 = Carrier b4 & Sum b3 = Sum b4;

:: RUSUB_3:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Linear_Combination of b2 holds
   ex b4 being Linear_Combination of b1 st
      Carrier b3 = Carrier b4 & Sum b3 = Sum b4;

:: RUSUB_3:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Linear_Combination of b1
      st Carrier b3 c= the carrier of b2
   holds ex b4 being Linear_Combination of b2 st
      Carrier b4 = Carrier b3 & Sum b4 = Sum b3;

:: RUSUB_3:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Basis of b1
for b3 being Element of the carrier of b1 holds
   b3 in Lin b2;

:: RUSUB_3:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b2
      st b3 is linearly-independent(b2)
   holds b3 is linearly-independent Element of bool the carrier of b1;

:: RUSUB_3:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b1
      st b3 is linearly-independent(b1) & b3 c= the carrier of b2
   holds b3 is linearly-independent Element of bool the carrier of b2;

:: RUSUB_3:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Basis of b2 holds
   ex b4 being Basis of b1 st
      b3 c= b4;

:: RUSUB_3:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
   st b2 is linearly-independent(b1)
for b3 being Element of the carrier of b1
   st b3 in b2
for b4 being Element of bool the carrier of b1
      st b4 = b2 \ {b3}
   holds not b3 in Lin b4;

:: RUSUB_3:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Basis of b1
for b3 being non empty Element of bool the carrier of b1
   st b3 misses b2
for b4 being Element of bool the carrier of b1
      st b4 = b2 \/ b3
   holds b4 is linearly-dependent(b1);

:: RUSUB_3:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b1
      st b3 c= the carrier of b2
   holds Lin b3 is Subspace of b2;

:: RUSUB_3:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Subspace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds Lin b3 = Lin b4;

:: RUSUB_3:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1
for b3 being set holds
      b3 in Lin {b2}
   iff
      ex b4 being Element of REAL st
         b3 = b4 * b2;

:: RUSUB_3:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being Element of the carrier of b1 holds
   b2 in Lin {b2};

:: RUSUB_3:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being set holds
      b4 in b2 + Lin {b3}
   iff
      ex b5 being Element of REAL st
         b4 = b2 + (b5 * b3);

:: RUSUB_3:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
for b4 being set holds
      b4 in Lin {b2,b3}
   iff
      ex b5, b6 being Element of REAL st
         b4 = (b5 * b2) + (b6 * b3);

:: RUSUB_3:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1 holds
b2 in Lin {b2,b3} & b3 in Lin {b2,b3};

:: RUSUB_3:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1
for b5 being set holds
      b5 in b2 + Lin {b3,b4}
   iff
      ex b6, b7 being Element of REAL st
         b5 = (b2 + (b6 * b3)) + (b7 * b4);

:: RUSUB_3:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Element of the carrier of b1
for b5 being set holds
      b5 in Lin {b2,b3,b4}
   iff
      ex b6, b7, b8 being Element of REAL st
         b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4);

:: RUSUB_3:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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};

:: RUSUB_3:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being set holds
      b6 in b2 + Lin {b3,b4,b5}
   iff
      ex b7, b8, b9 being Element of REAL st
         b6 = ((b2 + (b7 * b3)) + (b8 * b4)) + (b9 * b5);

:: RUSUB_3:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Element of the carrier of b1
      st b2 in Lin {b3} & b2 <> 0. b1
   holds Lin {b2} = Lin {b3};

:: RUSUB_3:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
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;

:: RUSUB_3:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2 being set holds
      b2 in (0). b1
   iff
      b2 = 0. b1;

:: RUSUB_3:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Subspace of b1
      st b2 is Subspace of b4
   holds b2 /\ b3 is Subspace of b4;

:: RUSUB_3:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Subspace of b1
      st b2 is Subspace of b3 & b2 is Subspace of b4
   holds b2 is Subspace of b3 /\ b4;

:: RUSUB_3:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Subspace of b1
      st b2 is Subspace of b4 & b3 is Subspace of b4
   holds b2 + b3 is Subspace of b4;

:: RUSUB_3:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3, b4 being Subspace of b1
      st b2 is Subspace of b3
   holds b2 is Subspace of b3 + b4;

:: RUSUB_3:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealUnitarySpace-like UNITSTR
for b2, b3 being Subspace of b1
for b4 being Element of the carrier of b1
      st b2 is Subspace of b3
   holds b4 + b2 c= b4 + b3;