Article RLSUB_2, MML version 4.99.1005

:: RLSUB_2:funcnot 1 => RLSUB_2:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Subspace of a1;
  func A2 + A3 -> strict Subspace of a1 means
    the carrier of it = {b1 + b2 where b1 is Element of the carrier of a1, b2 is Element of the carrier of a1: b1 in a2 & b2 in a3};
end;

:: RLSUB_2:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
for b4 being strict Subspace of b1 holds
      b4 = b2 + b3
   iff
      the carrier of b4 = {b5 + b6 where b5 is Element of the carrier of b1, b6 is Element of the carrier of b1: b5 in b2 & b6 in b3};

:: RLSUB_2:funcnot 2 => RLSUB_2:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Subspace of a1;
  func A2 /\ A3 -> strict Subspace of a1 means
    the carrier of it = (the carrier of a2) /\ the carrier of a3;
end;

:: RLSUB_2:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
for b4 being strict Subspace of b1 holds
      b4 = b2 /\ b3
   iff
      the carrier of b4 = (the carrier of b2) /\ the carrier of b3;

:: RLSUB_2:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
for b4 being set holds
      b4 in b2 + b3
   iff
      ex b5, b6 being Element of the carrier of b1 st
         b5 in b2 & b6 in b3 & b4 = b5 + b6;

:: RLSUB_2:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
for b4 being Element of the carrier of b1
      st (b4 in b2 or b4 in b3)
   holds b4 in b2 + b3;

:: RLSUB_2:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
for b4 being set holds
      b4 in b2 /\ b3
   iff
      b4 in b2 & b4 in b3;

:: RLSUB_2:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being strict Subspace of b1 holds
   b2 + b2 = b2;

:: RLSUB_2:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
b2 + b3 = b3 + b2;

:: RLSUB_2:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1 holds
b2 + (b3 + b4) = (b2 + b3) + b4;

:: RLSUB_2:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
b2 is Subspace of b2 + b3 & b3 is Subspace of b2 + b3;

:: RLSUB_2:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being strict Subspace of b1 holds
      b2 is Subspace of b3
   iff
      b2 + b3 = b3;

:: RLSUB_2:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being strict Subspace of b1 holds
   ((0). b1) + b2 = b2 & b2 + (0). b1 = b2;

:: RLSUB_2:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   ((0). b1) + (Omega). b1 = RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#) &
    ((Omega). b1) + (0). b1 = RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#);

:: RLSUB_2:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
   ((Omega). b1) + b2 = RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#) &
    b2 + (Omega). b1 = RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#);

:: RLSUB_2:th 16
theorem
for b1 being non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   ((Omega). b1) + (Omega). b1 = b1;

:: RLSUB_2:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being strict Subspace of b1 holds
   b2 /\ b2 = b2;

:: RLSUB_2:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
b2 /\ b3 = b3 /\ b2;

:: RLSUB_2:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1 holds
b2 /\ (b3 /\ b4) = (b2 /\ b3) /\ b4;

:: RLSUB_2:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
b2 /\ b3 is Subspace of b2 & b2 /\ b3 is Subspace of b3;

:: RLSUB_2:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being strict Subspace of b1 holds
      b3 is Subspace of b2
   iff
      b3 /\ b2 = b3;

:: RLSUB_2:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
   ((0). b1) /\ b2 = (0). b1 & b2 /\ (0). b1 = (0). b1;

:: RLSUB_2:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   ((0). b1) /\ (Omega). b1 = (0). b1 & ((Omega). b1) /\ (0). b1 = (0). b1;

:: RLSUB_2:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being strict Subspace of b1 holds
   ((Omega). b1) /\ b2 = b2 & b2 /\ (Omega). b1 = b2;

:: RLSUB_2:th 25
theorem
for b1 being non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   ((Omega). b1) /\ (Omega). b1 = b1;

:: RLSUB_2:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
b2 /\ b3 is Subspace of b2 + b3;

:: RLSUB_2:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being strict Subspace of b1 holds
   (b2 /\ b3) + b3 = b3;

:: RLSUB_2:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being strict Subspace of b1 holds
   b3 /\ (b3 + b2) = b3;

:: RLSUB_2:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1 holds
(b2 /\ b3) + (b3 /\ b4) is Subspace of b3 /\ (b2 + b4);

:: RLSUB_2:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1
      st b2 is Subspace of b3
   holds b3 /\ (b2 + b4) = (b2 /\ b3) + (b3 /\ b4);

:: RLSUB_2:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1 holds
b2 + (b3 /\ b4) is Subspace of (b3 + b2) /\ (b2 + b4);

:: RLSUB_2:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1
      st b2 is Subspace of b3
   holds b3 + (b2 /\ b4) = (b2 + b3) /\ (b3 + b4);

:: RLSUB_2:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1
      st b2 is strict Subspace of b3
   holds b2 + (b4 /\ b3) = (b2 + b4) /\ b3;

:: RLSUB_2:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being strict Subspace of b1 holds
   b2 + b3 = b3
iff
   b2 /\ b3 = b2;

:: RLSUB_2:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3, b4 being strict Subspace of b1
      st b2 is Subspace of b3
   holds b2 + b4 is Subspace of b3 + b4;

:: RLSUB_2:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
   ex b4 being Subspace of b1 st
      the carrier of b4 = (the carrier of b2) \/ the carrier of b3
iff
   (b2 is Subspace of b3 or b3 is Subspace of b2);

:: RLSUB_2:funcnot 3 => RLSUB_2:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func Subspaces A1 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 is strict Subspace of a1;
end;

:: RLSUB_2:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being set holds
      b2 = Subspaces b1
   iff
      for b3 being set holds
            b3 in b2
         iff
            b3 is strict Subspace of b1;

:: RLSUB_2:funcreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster Subspaces a1 -> non empty;
end;

:: RLSUB_2:th 39
theorem
for b1 being non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   b1 in Subspaces b1;

:: RLSUB_2:prednot 1 => RLSUB_2:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Subspace of a1;
  pred A1 is_the_direct_sum_of A2,A3 means
    RLSStruct(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1#) = a2 + a3 &
     a2 /\ a3 = (0). a1;
end;

:: RLSUB_2:dfs 4
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Subspace of a1;
To prove
     a1 is_the_direct_sum_of a2,a3
it is sufficient to prove
  thus RLSStruct(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1#) = a2 + a3 &
     a2 /\ a3 = (0). a1;

:: RLSUB_2:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
   b1 is_the_direct_sum_of b2,b3
iff
   RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#) = b2 + b3 &
    b2 /\ b3 = (0). b1;

:: RLSUB_2:modenot 1 => RLSUB_2:mode 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Subspace of a1;
  mode Linear_Compl of A2 -> Subspace of a1 means
    a1 is_the_direct_sum_of it,a2;
end;

:: RLSUB_2:dfs 5
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Subspace of a1;
To prove
     a3 is Linear_Compl of a2
it is sufficient to prove
  thus a1 is_the_direct_sum_of a3,a2;

:: RLSUB_2:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
   b3 is Linear_Compl of b2
iff
   b1 is_the_direct_sum_of b3,b2;

:: RLSUB_2:exreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Subspace of a1;
  cluster non empty left_complementable right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like zeroed Linear_Compl of a2;
end;

:: RLSUB_2:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
      st b1 is_the_direct_sum_of b2,b3
   holds b3 is Linear_Compl of b2;

:: RLSUB_2:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Compl of b2 holds
   b1 is_the_direct_sum_of b3,b2 & b1 is_the_direct_sum_of b2,b3;

:: RLSUB_2:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Compl of b2 holds
   b2 + b3 = RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#) &
    b3 + b2 = RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#);

:: RLSUB_2:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Compl of b2 holds
   b2 /\ b3 = (0). b1 & b3 /\ b2 = (0). b1;

:: RLSUB_2:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
      st b1 is_the_direct_sum_of b2,b3
   holds b1 is_the_direct_sum_of b3,b2;

:: RLSUB_2:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   b1 is_the_direct_sum_of (0). b1,(Omega). b1 & b1 is_the_direct_sum_of (Omega). b1,(0). b1;

:: RLSUB_2:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Compl of b2 holds
   b2 is Linear_Compl of b3;

:: RLSUB_2:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   (0). b1 is Linear_Compl of (Omega). b1 & (Omega). b1 is Linear_Compl of (0). b1;

:: RLSUB_2:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
for b4 being Coset of b2
for b5 being Coset of b3
      st b4 meets b5
   holds b4 /\ b5 is Coset of b2 /\ b3;

:: RLSUB_2:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
   b1 is_the_direct_sum_of b2,b3
iff
   for b4 being Coset of b2
   for b5 being Coset of b3 holds
      ex b6 being Element of the carrier of b1 st
         b4 /\ b5 = {b6};

:: RLSUB_2:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
   b2 + b3 = RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#)
iff
   for b4 being Element of the carrier of b1 holds
      ex b5, b6 being Element of the carrier of b1 st
         b5 in b2 & b6 in b3 & b4 = b5 + b6;

:: RLSUB_2:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
for b4, b5, b6, b7, b8 being Element of the carrier of b1
      st b1 is_the_direct_sum_of b2,b3 & b4 = b5 + b6 & b4 = b7 + b8 & b5 in b2 & b7 in b2 & b6 in b3 & b8 in b3
   holds b5 = b7 & b6 = b8;

:: RLSUB_2:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
      st b1 = b2 + b3 &
         (ex b4 being Element of the carrier of b1 st
            for b5, b6, b7, b8 being Element of the carrier of b1
                  st b4 = b5 + b6 & b4 = b7 + b8 & b5 in b2 & b7 in b2 & b6 in b3 & b8 in b3
               holds b5 = b7 & b6 = b8)
   holds b1 is_the_direct_sum_of b2,b3;

:: RLSUB_2:funcnot 4 => RLSUB_2:func 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Element of the carrier of a1;
  let a3, a4 be Subspace of a1;
  assume a1 is_the_direct_sum_of a3,a4;
  func A2 |--(A3,A4) -> Element of [:the carrier of a1,the carrier of a1:] means
    a2 = it `1 + (it `2) & it `1 in a3 & it `2 in a4;
end;

:: RLSUB_2:def 6
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 b1 is_the_direct_sum_of b3,b4
for b5 being Element of [:the carrier of b1,the carrier of b1:] holds
      b5 = b2 |--(b3,b4)
   iff
      b2 = b5 `1 + (b5 `2) & b5 `1 in b3 & b5 `2 in b4;

:: RLSUB_2:th 59
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
for b4 being Element of the carrier of b1
      st b1 is_the_direct_sum_of b2,b3
   holds (b4 |--(b2,b3)) `1 = (b4 |--(b3,b2)) `2;

:: RLSUB_2:th 60
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
for b4 being Element of the carrier of b1
      st b1 is_the_direct_sum_of b2,b3
   holds (b4 |--(b2,b3)) `2 = (b4 |--(b3,b2)) `1;

:: RLSUB_2:th 61
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Compl of b2
for b4 being Element of the carrier of b1
for b5 being Element of [:the carrier of b1,the carrier of b1:]
      st b5 `1 + (b5 `2) = b4 & b5 `1 in b2 & b5 `2 in b3
   holds b5 = b4 |--(b2,b3);

:: RLSUB_2:th 62
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Compl of b2
for b4 being Element of the carrier of b1 holds
   (b4 |--(b2,b3)) `1 + ((b4 |--(b2,b3)) `2) = b4;

:: RLSUB_2:th 63
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Compl of b2
for b4 being Element of the carrier of b1 holds
   (b4 |--(b2,b3)) `1 in b2 & (b4 |--(b2,b3)) `2 in b3;

:: RLSUB_2:th 64
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Compl of b2
for b4 being Element of the carrier of b1 holds
   (b4 |--(b2,b3)) `1 = (b4 |--(b3,b2)) `2;

:: RLSUB_2:th 65
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Linear_Compl of b2
for b4 being Element of the carrier of b1 holds
   (b4 |--(b2,b3)) `2 = (b4 |--(b3,b2)) `1;

:: RLSUB_2:funcnot 5 => RLSUB_2:func 5
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func SubJoin A1 -> Function-like quasi_total Relation of [:Subspaces a1,Subspaces a1:],Subspaces a1 means
    for b1, b2 being Element of Subspaces a1
    for b3, b4 being Subspace of a1
          st b1 = b3 & b2 = b4
       holds it .(b1,b2) = b3 + b4;
end;

:: RLSUB_2:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of [:Subspaces b1,Subspaces b1:],Subspaces b1 holds
      b2 = SubJoin b1
   iff
      for b3, b4 being Element of Subspaces b1
      for b5, b6 being Subspace of b1
            st b3 = b5 & b4 = b6
         holds b2 .(b3,b4) = b5 + b6;

:: RLSUB_2:funcnot 6 => RLSUB_2:func 6
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func SubMeet A1 -> Function-like quasi_total Relation of [:Subspaces a1,Subspaces a1:],Subspaces a1 means
    for b1, b2 being Element of Subspaces a1
    for b3, b4 being Subspace of a1
          st b1 = b3 & b2 = b4
       holds it .(b1,b2) = b3 /\ b4;
end;

:: RLSUB_2:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of [:Subspaces b1,Subspaces b1:],Subspaces b1 holds
      b2 = SubMeet b1
   iff
      for b3, b4 being Element of Subspaces b1
      for b5, b6 being Subspace of b1
            st b3 = b5 & b4 = b6
         holds b2 .(b3,b4) = b5 /\ b6;

:: RLSUB_2:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   LattStr(#Subspaces b1,SubJoin b1,SubMeet b1#) is non empty Lattice-like LattStr;

:: RLSUB_2:funcreg 2
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster LattStr(#Subspaces a1,SubJoin a1,SubMeet a1#) -> strict Lattice-like;
end;

:: RLSUB_2:th 71
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   LattStr(#Subspaces b1,SubJoin b1,SubMeet b1#) is lower-bounded;

:: RLSUB_2:th 72
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   LattStr(#Subspaces b1,SubJoin b1,SubMeet b1#) is upper-bounded;

:: RLSUB_2:th 73
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   LattStr(#Subspaces b1,SubJoin b1,SubMeet b1#) is non empty Lattice-like bounded LattStr;

:: RLSUB_2:th 74
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   LattStr(#Subspaces b1,SubJoin b1,SubMeet b1#) is modular;

:: RLSUB_2:th 75
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   LattStr(#Subspaces b1,SubJoin b1,SubMeet b1#) is complemented;

:: RLSUB_2:funcreg 3
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster LattStr(#Subspaces a1,SubJoin a1,SubMeet a1#) -> strict modular lower-bounded upper-bounded complemented;
end;

:: RLSUB_2:th 76
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being strict Subspace of b1
      st b2 is Subspace of b3
   holds b2 /\ b4 is Subspace of b3 /\ b4;

:: RLSUB_2:th 78
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
   b2 = b3 + b4
iff
   b3 = b2 - b4;

:: RLSUB_2:th 79
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being strict Subspace of b1
      st for b3 being Element of the carrier of b1 holds
           b3 in b2
   holds b2 = RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#);

:: RLSUB_2:th 80
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1
for b3 being Element of the carrier of b1 holds
   ex b4 being Coset of b2 st
      b3 in b4;

:: RLSUB_2:th 84
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Element of the carrier of b1
      st for b3 being Element of the carrier of b1 holds
           b3 "/\" b2 = b2
   holds b2 = Bottom b1;

:: RLSUB_2:th 85
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Element of the carrier of b1
      st for b3 being Element of the carrier of b1 holds
           b3 "\/" b2 = b2
   holds b2 = Top b1;