Article RLSUB_1, MML version 4.99.1005

:: RLSUB_1:attrnot 1 => RLSUB_1:attr 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is linearly-closed means
    (for b1, b2 being Element of the carrier of a1
           st b1 in a2 & b2 in a2
        holds b1 + b2 in a2) &
     (for b1 being Element of REAL
     for b2 being Element of the carrier of a1
           st b2 in a2
        holds b1 * b2 in a2);
end;

:: RLSUB_1:dfs 1
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is linearly-closed
it is sufficient to prove
  thus (for b1, b2 being Element of the carrier of a1
           st b1 in a2 & b2 in a2
        holds b1 + b2 in a2) &
     (for b1 being Element of REAL
     for b2 being Element of the carrier of a1
           st b2 in a2
        holds b1 * b2 in a2);

:: RLSUB_1:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is linearly-closed(b1)
   iff
      (for b3, b4 being Element of the carrier of b1
             st b3 in b2 & b4 in b2
          holds b3 + b4 in b2) &
       (for b3 being Element of REAL
       for b4 being Element of the carrier of b1
             st b4 in b2
          holds b3 * b4 in b2);

:: RLSUB_1:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
      st b2 <> {} & b2 is linearly-closed(b1)
   holds 0. b1 in b2;

:: RLSUB_1:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
   st b2 is linearly-closed(b1)
for b3 being Element of the carrier of b1
      st b3 in b2
   holds - b3 in b2;

:: RLSUB_1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
   st b2 is linearly-closed(b1)
for b3, b4 being Element of the carrier of b1
      st b3 in b2 & b4 in b2
   holds b3 - b4 in b2;

:: RLSUB_1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   {0. b1} is linearly-closed(b1);

:: RLSUB_1:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
      st the carrier of b1 = b2
   holds b2 is linearly-closed(b1);

:: RLSUB_1: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 bool the carrier of b1
      st b2 is linearly-closed(b1) &
         b3 is linearly-closed(b1) &
         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}
   holds b4 is linearly-closed(b1);

:: RLSUB_1:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is linearly-closed(b1) & b3 is linearly-closed(b1)
   holds b2 /\ b3 is linearly-closed(b1);

:: RLSUB_1:modenot 1 => RLSUB_1:mode 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  mode Subspace of A1 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct means
    the carrier of it c= the carrier of a1 &
     0. it = 0. a1 &
     the addF of it = (the addF of a1) || the carrier of it &
     the Mult of it = (the Mult of a1) | [:REAL,the carrier of it:];
end;

:: RLSUB_1:dfs 2
definiens
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
To prove
     a2 is Subspace of a1
it is sufficient to prove
  thus the carrier of a2 c= the carrier of a1 &
     0. a2 = 0. a1 &
     the addF of a2 = (the addF of a1) || the carrier of a2 &
     the Mult of a2 = (the Mult of a1) | [:REAL,the carrier of a2:];

:: RLSUB_1:def 2
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   b2 is Subspace of b1
iff
   the carrier of b2 c= the carrier of b1 &
    0. b2 = 0. b1 &
    the addF of b2 = (the addF of b1) || the carrier of b2 &
    the Mult of b2 = (the Mult of b1) | [:REAL,the carrier of b2:];

:: RLSUB_1:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being set
for b3, b4 being Subspace of b1
      st b2 in b3 & b3 is Subspace of b4
   holds b2 in b4;

:: RLSUB_1:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being set
for b3 being Subspace of b1
      st b2 in b3
   holds b2 in b1;

:: RLSUB_1:th 18
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 b2 holds
   b3 is Element of the carrier of b1;

:: RLSUB_1:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
   0. b2 = 0. b1;

:: RLSUB_1: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
0. b2 = 0. b3;

:: RLSUB_1: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
for b4 being Subspace of b1
for b5, b6 being Element of the carrier of b4
      st b5 = b2 & b6 = b3
   holds b5 + b6 = b2 + b3;

:: RLSUB_1:th 22
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 being Element of REAL
for b4 being Subspace of b1
for b5 being Element of the carrier of b4
      st b5 = b2
   holds b3 * b5 = b3 * b2;

:: RLSUB_1:th 23
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 being Subspace of b1
for b4 being Element of the carrier of b3
      st b4 = b2
   holds - b2 = - b4;

:: RLSUB_1:th 24
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
for b5, b6 being Element of the carrier of b4
      st b5 = b2 & b6 = b3
   holds b5 - b6 = b2 - b3;

:: RLSUB_1:th 25
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 in b2;

:: RLSUB_1: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
0. b2 in b3;

:: RLSUB_1:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
   0. b2 in b1;

:: RLSUB_1:th 28
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 b4 & b3 in b4
   holds b2 + b3 in b4;

:: RLSUB_1:th 29
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 being Element of REAL
for b4 being Subspace of b1
      st b2 in b4
   holds b3 * b2 in b4;

:: RLSUB_1:th 30
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 being Subspace of b1
      st b2 in b3
   holds - b2 in b3;

:: RLSUB_1:th 31
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 b4 & b3 in b4
   holds b2 - b3 in b4;

:: RLSUB_1:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty set
for b4 being Element of b3
for b5 being Function-like quasi_total Relation of [:b3,b3:],b3
for b6 being Function-like quasi_total Relation of [:REAL,b3:],b3
      st b2 = b3 &
         b4 = 0. b1 &
         b5 = (the addF of b1) || b2 &
         b6 = (the Mult of b1) | [:REAL,b2:]
   holds RLSStruct(#b3,b4,b5,b6#) is Subspace of b1;

:: RLSUB_1:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   b1 is Subspace of b1;

:: RLSUB_1:th 34
theorem
for b1, b2 being non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
      st b1 is Subspace of b2 & b2 is Subspace of b1
   holds b1 = b2;

:: RLSUB_1:th 35
theorem
for b1, b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
      st b1 is Subspace of b2 & b2 is Subspace of b3
   holds b1 is Subspace of b3;

:: RLSUB_1: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
      st the carrier of b2 c= the carrier of b3
   holds b2 is Subspace of b3;

:: RLSUB_1:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1
      st for b4 being Element of the carrier of b1
              st b4 in b2
           holds b4 in b3
   holds b2 is Subspace of b3;

:: RLSUB_1:exreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed RealLinearSpace-like zeroed Subspace of a1;
end;

:: RLSUB_1:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being strict Subspace of b1
      st the carrier of b2 = the carrier of b3
   holds b2 = b3;

:: RLSUB_1:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being strict Subspace of b1
      st for b4 being Element of the carrier of b1 holds
              b4 in b2
           iff
              b4 in b3
   holds b2 = b3;

:: RLSUB_1:th 40
theorem
for b1 being non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being strict Subspace of b1
      st the carrier of b2 = the carrier of b1
   holds b2 = b1;

:: RLSUB_1:th 41
theorem
for b1 being non empty right_complementable strict 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
           iff
              b3 in b1
   holds b2 = b1;

:: RLSUB_1:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Subspace of b1
      st the carrier of b3 = b2
   holds b2 is linearly-closed(b1);

:: RLSUB_1:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
      st b2 <> {} & b2 is linearly-closed(b1)
   holds ex b3 being strict Subspace of b1 st
      b2 = the carrier of b3;

:: RLSUB_1:funcnot 1 => RLSUB_1:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func (0). A1 -> strict Subspace of a1 means
    the carrier of it = {0. a1};
end;

:: RLSUB_1:def 3
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 = (0). b1
   iff
      the carrier of b2 = {0. b1};

:: RLSUB_1:funcnot 2 => RLSUB_1:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func (Omega). A1 -> strict Subspace of a1 equals
    RLSStruct(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1#);
end;

:: RLSUB_1:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   (Omega). b1 = RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#);

:: RLSUB_1:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
   (0). b2 = (0). b1;

:: RLSUB_1:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Subspace of b1 holds
(0). b2 = (0). b3;

:: RLSUB_1:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
   (0). b2 is Subspace of b1;

:: RLSUB_1:th 51
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 is Subspace of b2;

:: RLSUB_1: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
(0). b2 is Subspace of b3;

:: RLSUB_1:th 54
theorem
for b1 being non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   b1 is Subspace of (Omega). b1;

:: RLSUB_1:funcnot 3 => RLSUB_1:func 3
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 be Subspace of a1;
  func A2 + A3 -> Element of bool the carrier of a1 equals
    {a2 + b1 where b1 is Element of the carrier of a1: b1 in a3};
end;

:: RLSUB_1:def 5
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 being Subspace of b1 holds
   b2 + b3 = {b2 + b4 where b4 is Element of the carrier of b1: b4 in b3};

:: RLSUB_1:modenot 2 => RLSUB_1:mode 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Subspace of a1;
  mode Coset of A2 -> Element of bool the carrier of a1 means
    ex b1 being Element of the carrier of a1 st
       it = b1 + a2;
end;

:: RLSUB_1:dfs 6
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Subspace of a1;
  let a3 be Element of bool the carrier of a1;
To prove
     a3 is Coset of a2
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       a3 = b1 + a2;

:: RLSUB_1:def 6
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 bool the carrier of b1 holds
      b3 is Coset of b2
   iff
      ex b4 being Element of the carrier of b1 st
         b3 = b4 + b2;

:: RLSUB_1:th 58
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 being Subspace of b1 holds
      0. b1 in b2 + b3
   iff
      b2 in b3;

:: RLSUB_1:th 59
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 being Subspace of b1 holds
   b2 in b2 + b3;

:: RLSUB_1:th 60
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 = the carrier of b2;

:: RLSUB_1:th 61
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 + (0). b1 = {b2};

:: RLSUB_1:th 62
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 + (Omega). b1 = the carrier of b1;

:: RLSUB_1:th 63
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 being Subspace of b1 holds
      0. b1 in b2 + b3
   iff
      b2 + b3 = the carrier of b3;

:: RLSUB_1:th 64
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 being Subspace of b1 holds
      b2 in b3
   iff
      b2 + b3 = the carrier of b3;

:: RLSUB_1:th 65
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 being Element of REAL
for b4 being Subspace of b1
      st b2 in b4
   holds (b3 * b2) + b4 = the carrier of b4;

:: RLSUB_1:th 66
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 being Element of REAL
for b4 being Subspace of b1
      st b3 <> 0 & (b3 * b2) + b4 = the carrier of b4
   holds b2 in b4;

:: RLSUB_1:th 67
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 being Subspace of b1 holds
      b2 in b3
   iff
      (- b2) + b3 = the carrier of b3;

:: RLSUB_1:th 68
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 holds
      b2 in b4
   iff
      b3 + b4 = (b3 + b2) + b4;

:: RLSUB_1:th 69
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 holds
      b2 in b4
   iff
      b3 + b4 = (b3 - b2) + b4;

:: RLSUB_1:th 70
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 holds
      b2 in b3 + b4
   iff
      b3 + b4 = b2 + b4;

:: RLSUB_1:th 71
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 being Subspace of b1 holds
      b2 + b3 = (- b2) + b3
   iff
      b2 in b3;

:: RLSUB_1:th 72
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 Subspace of b1
      st b2 in b3 + b5 & b2 in b4 + b5
   holds b3 + b5 = b4 + b5;

:: RLSUB_1:th 73
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 & b2 in (- b3) + b4
   holds b3 in b4;

:: RLSUB_1:th 74
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 being Element of REAL
for b4 being Subspace of b1
      st b3 <> 1 & b3 * b2 in b2 + b4
   holds b2 in b4;

:: RLSUB_1:th 75
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 being Element of REAL
for b4 being Subspace of b1
      st b2 in b4
   holds b3 * b2 in b2 + b4;

:: RLSUB_1:th 76
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 being Subspace of b1 holds
      - b2 in b2 + b3
   iff
      b2 in b3;

:: RLSUB_1:th 77
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 holds
      b2 + b3 in b3 + b4
   iff
      b2 in b4;

:: RLSUB_1:th 78
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 holds
      b2 - b3 in b2 + b4
   iff
      b3 in b4;

:: RLSUB_1:th 79
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 holds
      b2 in b3 + b4
   iff
      ex b5 being Element of the carrier of b1 st
         b5 in b4 & b2 = b3 + b5;

:: RLSUB_1:th 80
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 holds
      b2 in b3 + b4
   iff
      ex b5 being Element of the carrier of b1 st
         b5 in b4 & b2 = b3 - b5;

:: RLSUB_1:th 81
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 holds
      ex b5 being Element of the carrier of b1 st
         b2 in b5 + b4 & b3 in b5 + b4
   iff
      b2 - b3 in b4;

:: RLSUB_1:th 82
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 + b4 = b3 + b4
   holds ex b5 being Element of the carrier of b1 st
      b5 in b4 & b2 + b5 = b3;

:: RLSUB_1:th 83
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 + b4 = b3 + b4
   holds ex b5 being Element of the carrier of b1 st
      b5 in b4 & b2 - b5 = b3;

:: RLSUB_1:th 84
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 strict Subspace of b1 holds
   b2 + b3 = b2 + b4
iff
   b3 = b4;

:: RLSUB_1:th 85
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, b5 being strict Subspace of b1
      st b2 + b4 = b3 + b5
   holds b4 = b5;

:: RLSUB_1:th 86
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 Coset of b2 holds
      b3 is linearly-closed(b1)
   iff
      b3 = the carrier of b2;

:: RLSUB_1:th 87
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being strict Subspace of b1
for b4 being Coset of b2
for b5 being Coset of b3
      st b4 = b5
   holds b2 = b3;

:: RLSUB_1:th 88
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} is Coset of (0). b1;

:: RLSUB_1:th 89
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is Coset of (0). b1
   holds ex b3 being Element of the carrier of b1 st
      b2 = {b3};

:: RLSUB_1:th 90
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Subspace of b1 holds
   the carrier of b2 is Coset of b2;

:: RLSUB_1:th 91
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   the carrier of b1 is Coset of (Omega). b1;

:: RLSUB_1:th 92
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is Coset of (Omega). b1
   holds b2 = the carrier of b1;

:: RLSUB_1:th 93
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 Coset of b2 holds
      0. b1 in b3
   iff
      b3 = the carrier of b2;

:: RLSUB_1:th 94
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 being Subspace of b1
for b4 being Coset of b3 holds
      b2 in b4
   iff
      b4 = b2 + b3;

:: RLSUB_1:th 95
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
for b5 being Coset of b4
      st b2 in b5 & b3 in b5
   holds ex b6 being Element of the carrier of b1 st
      b6 in b4 & b2 + b6 = b3;

:: RLSUB_1:th 96
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
for b5 being Coset of b4
      st b2 in b5 & b3 in b5
   holds ex b6 being Element of the carrier of b1 st
      b6 in b4 & b2 - b6 = b3;

:: RLSUB_1:th 97
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 holds
      ex b5 being Coset of b4 st
         b2 in b5 & b3 in b5
   iff
      b2 - b3 in b4;

:: RLSUB_1:th 98
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 being Subspace of b1
for b4, b5 being Coset of b3
      st b2 in b4 & b2 in b5
   holds b4 = b5;