Article RLTOPSP1, MML version 4.99.1005

:: RLTOPSP1:th 1
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b3 in b2
   holds b4 * b3 in b4 * b2;

:: RLTOPSP1:exreg 1
registration
  cluster non empty ext-real complex real Element of REAL;
end;

:: RLTOPSP1:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1
   st b3 is_a_cover_of b2
for b4 being Element of the carrier of b1
      st b4 in b2
   holds ex b5 being Element of bool the carrier of b1 st
      b4 in b5 & b5 in b3;

:: RLTOPSP1:th 3
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of the carrier of b1
      st b4 in b2 & b5 in b3
   holds b4 + b5 in b2 + b3;

:: RLTOPSP1:th 4
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of bool bool the carrier of b1
      st b4 = {b5 + b3 where b5 is Element of the carrier of b1: b5 in b2}
   holds b2 + b3 = union b4;

:: RLTOPSP1:th 5
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of bool the carrier of b1 holds
   (0. b1) + b2 = b2;

:: RLTOPSP1:th 6
theorem
for b1 being non empty add-associative addLoopStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
   (b2 + b3) + b4 = b2 + (b3 + b4);

:: RLTOPSP1:th 7
theorem
for b1 being non empty add-associative addLoopStr
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4);

:: RLTOPSP1:th 8
theorem
for b1 being non empty addLoopStr
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
      st b2 c= b3
   holds b4 + b2 c= b4 + b3;

:: RLTOPSP1:th 9
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st b3 in b2
   holds 0. b1 in (- b3) + b2;

:: RLTOPSP1:th 10
theorem
for b1 being non empty addLoopStr
for b2, b3, b4 being Element of bool the carrier of b1
      st b2 c= b3
   holds b2 + b4 c= b3 + b4;

:: RLTOPSP1:th 11
theorem
for b1 being non empty addLoopStr
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b2 c= b4 & b3 c= b5
   holds b2 + b3 c= b4 + b5;

:: RLTOPSP1:th 12
theorem
for b1 being non empty right_zeroed addLoopStr
for b2, b3 being Element of bool the carrier of b1
      st 0. b1 in b3
   holds b2 c= b2 + b3;

:: RLTOPSP1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of REAL holds
   b2 * {0. b1} = {0. b1};

:: RLTOPSP1:th 14
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 Element of REAL
      st 0. b1 in b3 * b2
   holds 0. b1 in b2;

:: RLTOPSP1:th 15
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
for b4 being non empty Element of REAL holds
   (b4 * b2) /\ (b4 * b3) = b4 * (b2 /\ b3);

:: RLTOPSP1:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being a_neighborhood of b2
for b4 being Element of bool the carrier of b1
      st b3 c= b4
   holds b4 is a_neighborhood of b2;

:: RLTOPSP1:attrnot 1 => CONVEX1:attr 1
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is convex means
    for b1, b2 being Element of the carrier of a1
    for b3 being Element of REAL
          st 0 <= b3 & b3 <= 1 & b1 in a2 & b2 in a2
       holds (b3 * b1) + ((1 - b3) * b2) in a2;
end;

:: RLTOPSP1: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 convex
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
    for b3 being Element of REAL
          st 0 <= b3 & b3 <= 1 & b1 in a2 & b2 in a2
       holds (b3 * b1) + ((1 - b3) * b2) in a2;

:: RLTOPSP1: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 convex(b1)
   iff
      for b3, b4 being Element of the carrier of b1
      for b5 being Element of REAL
            st 0 <= b5 & b5 <= 1 & b3 in b2 & b4 in b2
         holds (b5 * b3) + ((1 - b5) * b4) in b2;

:: RLTOPSP1:funcreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be empty Element of bool the carrier of a1;
  cluster conv a2 -> empty convex;
end;

:: RLTOPSP1:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being convex Element of bool the carrier of b1 holds
   conv b2 = b2;

:: RLTOPSP1:th 19
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 Element of REAL holds
   b3 * conv b2 = conv (b3 * b2);

:: RLTOPSP1:th 20
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 c= b3
   holds Convex-Family b3 c= Convex-Family b2;

:: RLTOPSP1:th 21
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 c= b3
   holds conv b2 c= conv b3;

:: RLTOPSP1:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being convex Element of bool the carrier of b1
for b3 being Element of REAL
      st 0 <= b3 & b3 <= 1 & 0. b1 in b2
   holds b3 * b2 c= b2;

:: RLTOPSP1:funcnot 1 => RLTOPSP1:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Element of the carrier of a1;
  func LSeg(A2,A3) -> Element of bool the carrier of a1 equals
    {((1 - b1) * a2) + (b1 * a3) where b1 is Element of REAL: 0 <= b1 & b1 <= 1};
end;

:: RLTOPSP1:def 2
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
LSeg(b2,b3) = {((1 - b4) * b2) + (b4 * b3) where b4 is Element of REAL: 0 <= b4 & b4 <= 1};

:: RLTOPSP1:funcreg 2
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be Element of the carrier of a1;
  cluster LSeg(a2,a3) -> non empty convex;
end;

:: RLTOPSP1:th 23
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 convex(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st b3 in b2 & b4 in b2
         holds LSeg(b3,b4) c= b2;

:: RLTOPSP1:attrnot 2 => RLTOPSP1:attr 1
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is convex-membered means
    for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is convex(a1);
end;

:: RLTOPSP1:dfs 3
definiens
  let a1 be non empty RLSStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is convex-membered
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is convex(a1);

:: RLTOPSP1:def 3
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is convex-membered(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 in b2
         holds b3 is convex(b1);

:: RLTOPSP1:exreg 2
registration
  let a1 be non empty RLSStruct;
  cluster non empty convex-membered Element of bool bool the carrier of a1;
end;

:: RLTOPSP1:th 24
theorem
for b1 being non empty RLSStruct
for b2 being convex-membered Element of bool bool the carrier of b1 holds
   meet b2 is convex(b1);

:: RLTOPSP1:funcnot 2 => RLTOPSP1:func 2
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  func - A2 -> Element of bool the carrier of a1 equals
    (- 1) * a2;
end;

:: RLTOPSP1:def 4
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1 holds
   - b2 = (- 1) * b2;

:: RLTOPSP1:th 25
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
for b4 being Element of the carrier of b1 holds
      b4 + b2 meets b3
   iff
      b4 in b3 + - b2;

:: RLTOPSP1:attrnot 3 => RLTOPSP1:attr 2
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is symmetric means
    a2 = - a2;
end;

:: RLTOPSP1:dfs 5
definiens
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is symmetric
it is sufficient to prove
  thus a2 = - a2;

:: RLTOPSP1:def 5
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is symmetric(b1)
   iff
      b2 = - b2;

:: RLTOPSP1:exreg 3
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster non empty symmetric Element of bool the carrier of a1;
end;

:: RLTOPSP1:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being symmetric Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st b3 in b2
   holds - b3 in b2;

:: RLTOPSP1:attrnot 4 => RLTOPSP1:attr 3
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is circled means
    for b1 being Element of REAL
          st abs b1 <= 1
       holds b1 * a2 c= a2;
end;

:: RLTOPSP1:dfs 6
definiens
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is circled
it is sufficient to prove
  thus for b1 being Element of REAL
          st abs b1 <= 1
       holds b1 * a2 c= a2;

:: RLTOPSP1:def 6
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is circled(b1)
   iff
      for b3 being Element of REAL
            st abs b3 <= 1
         holds b3 * b2 c= b2;

:: RLTOPSP1:funcreg 3
registration
  let a1 be non empty RLSStruct;
  cluster {} a1 -> circled;
end;

:: RLTOPSP1:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   {0. b1} is circled(b1);

:: RLTOPSP1:exreg 4
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster non empty circled Element of bool the carrier of a1;
end;

:: RLTOPSP1:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty circled Element of bool the carrier of b1 holds
   0. b1 in b2;

:: RLTOPSP1:funcreg 4
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2, a3 be circled Element of bool the carrier of a1;
  cluster a2 + a3 -> circled;
end;

:: RLTOPSP1:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being circled Element of bool the carrier of b1
for b3 being Element of REAL
      st abs b3 = 1
   holds b3 * b2 = b2;

:: RLTOPSP1:condreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster circled -> symmetric (Element of bool the carrier of a1);
end;

:: RLTOPSP1:funcreg 5
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be circled Element of bool the carrier of a1;
  cluster conv a2 -> convex circled;
end;

:: RLTOPSP1:attrnot 5 => RLTOPSP1:attr 4
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is circled-membered means
    for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is circled(a1);
end;

:: RLTOPSP1:dfs 7
definiens
  let a1 be non empty RLSStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is circled-membered
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 in a2
       holds b1 is circled(a1);

:: RLTOPSP1:def 7
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is circled-membered(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 in b2
         holds b3 is circled(b1);

:: RLTOPSP1:exreg 5
registration
  let a1 be non empty RLSStruct;
  cluster non empty circled-membered Element of bool bool the carrier of a1;
end;

:: RLTOPSP1:th 30
theorem
for b1 being non empty RLSStruct
for b2 being circled-membered Element of bool bool the carrier of b1 holds
   union b2 is circled(b1);

:: RLTOPSP1:th 31
theorem
for b1 being non empty RLSStruct
for b2 being circled-membered Element of bool bool the carrier of b1 holds
   meet b2 is circled(b1);

:: RLTOPSP1:structnot 1 => RLTOPSP1:struct 1
definition
  struct(RLSStructTopStruct) RLTopStruct(#
    carrier -> set,
    ZeroF -> Element of the carrier of it,
    addF -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it,
    Mult -> Function-like quasi_total Relation of [:REAL,the carrier of it:],the carrier of it,
    topology -> Element of bool bool the carrier of it
  #);
end;

:: RLTOPSP1:attrnot 6 => RLTOPSP1:attr 5
definition
  let a1 be RLTopStruct;
  attr a1 is strict;
end;

:: RLTOPSP1:exreg 6
registration
  cluster strict RLTopStruct;
end;

:: RLTOPSP1:aggrnot 1 => RLTOPSP1:aggr 1
definition
  let a1 be set;
  let a2 be Element of a1;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a4 be Function-like quasi_total Relation of [:REAL,a1:],a1;
  let a5 be Element of bool bool a1;
  aggr RLTopStruct(#a1,a2,a3,a4,a5#) -> strict RLTopStruct;
end;

:: RLTOPSP1:funcreg 6
registration
  let a1 be non empty set;
  let a2 be Element of a1;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a4 be Function-like quasi_total Relation of [:REAL,a1:],a1;
  let a5 be Element of bool bool a1;
  cluster RLTopStruct(#a1,a2,a3,a4,a5#) -> non empty strict;
end;

:: RLTOPSP1:exreg 7
registration
  cluster non empty strict RLTopStruct;
end;

:: RLTOPSP1:attrnot 7 => RLTOPSP1:attr 6
definition
  let a1 be non empty RLTopStruct;
  attr a1 is add-continuous means
    for b1, b2 being Element of the carrier of a1
    for b3 being Element of bool the carrier of a1
          st b3 is open(a1) & b1 + b2 in b3
       holds ex b4, b5 being Element of bool the carrier of a1 st
          b4 is open(a1) & b5 is open(a1) & b1 in b4 & b2 in b5 & b4 + b5 c= b3;
end;

:: RLTOPSP1:dfs 8
definiens
  let a1 be non empty RLTopStruct;
To prove
     a1 is add-continuous
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
    for b3 being Element of bool the carrier of a1
          st b3 is open(a1) & b1 + b2 in b3
       holds ex b4, b5 being Element of bool the carrier of a1 st
          b4 is open(a1) & b5 is open(a1) & b1 in b4 & b2 in b5 & b4 + b5 c= b3;

:: RLTOPSP1:def 8
theorem
for b1 being non empty RLTopStruct holds
      b1 is add-continuous
   iff
      for b2, b3 being Element of the carrier of b1
      for b4 being Element of bool the carrier of b1
            st b4 is open(b1) & b2 + b3 in b4
         holds ex b5, b6 being Element of bool the carrier of b1 st
            b5 is open(b1) & b6 is open(b1) & b2 in b5 & b3 in b6 & b5 + b6 c= b4;

:: RLTOPSP1:attrnot 8 => RLTOPSP1:attr 7
definition
  let a1 be non empty RLTopStruct;
  attr a1 is Mult-continuous means
    for b1 being Element of REAL
    for b2 being Element of the carrier of a1
    for b3 being Element of bool the carrier of a1
          st b3 is open(a1) & b1 * b2 in b3
       holds ex b4 being positive Element of REAL st
          ex b5 being Element of bool the carrier of a1 st
             b5 is open(a1) &
              b2 in b5 &
              (for b6 being Element of REAL
                    st abs (b6 - b1) < b4
                 holds b6 * b5 c= b3);
end;

:: RLTOPSP1:dfs 9
definiens
  let a1 be non empty RLTopStruct;
To prove
     a1 is Mult-continuous
it is sufficient to prove
  thus for b1 being Element of REAL
    for b2 being Element of the carrier of a1
    for b3 being Element of bool the carrier of a1
          st b3 is open(a1) & b1 * b2 in b3
       holds ex b4 being positive Element of REAL st
          ex b5 being Element of bool the carrier of a1 st
             b5 is open(a1) &
              b2 in b5 &
              (for b6 being Element of REAL
                    st abs (b6 - b1) < b4
                 holds b6 * b5 c= b3);

:: RLTOPSP1:def 9
theorem
for b1 being non empty RLTopStruct holds
      b1 is Mult-continuous
   iff
      for b2 being Element of REAL
      for b3 being Element of the carrier of b1
      for b4 being Element of bool the carrier of b1
            st b4 is open(b1) & b2 * b3 in b4
         holds ex b5 being positive Element of REAL st
            ex b6 being Element of bool the carrier of b1 st
               b6 is open(b1) &
                b3 in b6 &
                (for b7 being Element of REAL
                      st abs (b7 - b2) < b5
                   holds b7 * b6 c= b4);

:: RLTOPSP1:exreg 8
registration
  cluster non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict add-continuous Mult-continuous RLTopStruct;
end;

:: RLTOPSP1:modenot 1
definition
  mode LinearTopSpace is non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
end;

:: RLTOPSP1:th 32
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2, b3 being Element of the carrier of b1
for b4 being a_neighborhood of b2 + b3 holds
   ex b5 being a_neighborhood of b2 st
      ex b6 being a_neighborhood of b3 st
         b5 + b6 c= b4;

:: RLTOPSP1:th 33
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of REAL
for b3 being Element of the carrier of b1
for b4 being a_neighborhood of b2 * b3 holds
   ex b5 being positive Element of REAL st
      ex b6 being a_neighborhood of b3 st
         for b7 being Element of REAL
               st abs (b7 - b2) < b5
            holds b7 * b6 c= b4;

:: RLTOPSP1:funcnot 3 => RLTOPSP1:func 3
definition
  let a1 be non empty RLTopStruct;
  let a2 be Element of the carrier of a1;
  func transl(A2,A1) -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = a2 + b1;
end;

:: RLTOPSP1:def 10
theorem
for b1 being non empty RLTopStruct
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
      b3 = transl(b2,b1)
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = b2 + b4;

:: RLTOPSP1:th 34
theorem
for b1 being non empty RLTopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
   (transl(b2,b1)) .: b3 = b2 + b3;

:: RLTOPSP1:th 35
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of the carrier of b1 holds
   rng transl(b2,b1) = [#] b1;

:: RLTOPSP1:th 36
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of the carrier of b1 holds
   (transl(b2,b1)) /" = transl(- b2,b1);

:: RLTOPSP1:funcreg 7
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be Element of the carrier of a1;
  cluster transl(a2,a1) -> Function-like quasi_total being_homeomorphism;
end;

:: RLTOPSP1:funcreg 8
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be open Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  cluster a3 + a2 -> open;
end;

:: RLTOPSP1:funcreg 9
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be open Element of bool the carrier of a1;
  let a3 be Element of bool the carrier of a1;
  cluster a3 + a2 -> open;
end;

:: RLTOPSP1:funcreg 10
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be closed Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  cluster a3 + a2 -> closed;
end;

:: RLTOPSP1:th 37
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b2 + b3 c= b4
   holds (Int b2) + Int b3 c= Int b4;

:: RLTOPSP1:th 38
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
   b2 + Int b3 = Int (b2 + b3);

:: RLTOPSP1:th 39
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
   b2 + Cl b3 = Cl (b2 + b3);

:: RLTOPSP1:th 40
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2, b3 being Element of the carrier of b1
for b4 being a_neighborhood of b2 holds
   b3 + b4 is a_neighborhood of b3 + b2;

:: RLTOPSP1:th 41
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of the carrier of b1
for b3 being a_neighborhood of b2 holds
   (- b2) + b3 is a_neighborhood of 0. b1;

:: RLTOPSP1:modenot 2
definition
  let a1 be non empty RLTopStruct;
  mode local_base of a1 is basis of 0. a1;
end;

:: RLTOPSP1:attrnot 9 => RLTOPSP1:attr 8
definition
  let a1 be non empty RLTopStruct;
  attr a1 is locally-convex means
    ex b1 being basis of 0. a1 st
       b1 is convex-membered(a1);
end;

:: RLTOPSP1:dfs 11
definiens
  let a1 be non empty RLTopStruct;
To prove
     a1 is locally-convex
it is sufficient to prove
  thus ex b1 being basis of 0. a1 st
       b1 is convex-membered(a1);

:: RLTOPSP1:def 11
theorem
for b1 being non empty RLTopStruct holds
      b1 is locally-convex
   iff
      ex b2 being basis of 0. b1 st
         b2 is convex-membered(b1);

:: RLTOPSP1:attrnot 10 => RLTOPSP1:attr 9
definition
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is bounded means
    for b1 being a_neighborhood of 0. a1 holds
       ex b2 being Element of REAL st
          0 < b2 &
           (for b3 being Element of REAL
                 st b2 < b3
              holds a2 c= b3 * b1);
end;

:: RLTOPSP1:dfs 12
definiens
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is bounded
it is sufficient to prove
  thus for b1 being a_neighborhood of 0. a1 holds
       ex b2 being Element of REAL st
          0 < b2 &
           (for b3 being Element of REAL
                 st b2 < b3
              holds a2 c= b3 * b1);

:: RLTOPSP1:def 12
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is bounded(b1)
   iff
      for b3 being a_neighborhood of 0. b1 holds
         ex b4 being Element of REAL st
            0 < b4 &
             (for b5 being Element of REAL
                   st b4 < b5
                holds b2 c= b5 * b3);

:: RLTOPSP1:funcreg 11
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  cluster {} a1 -> bounded;
end;

:: RLTOPSP1:exreg 9
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  cluster bounded Element of bool the carrier of a1;
end;

:: RLTOPSP1:th 42
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2, b3 being bounded Element of bool the carrier of b1 holds
b2 \/ b3 is bounded(b1);

:: RLTOPSP1:th 43
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being bounded Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 c= b2
   holds b3 is bounded(b1);

:: RLTOPSP1:th 44
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is finite &
         b2 = {b3 where b3 is bounded Element of bool the carrier of b1: TRUE}
   holds union b2 is bounded(b1);

:: RLTOPSP1:th 45
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 = {b3 where b3 is a_neighborhood of 0. b1: TRUE}
   holds b2 is basis of 0. b1;

:: RLTOPSP1:th 46
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being basis of 0. b1
for b3 being Element of bool bool the carrier of b1
      st b3 = {b4 + b5 where b4 is Element of the carrier of b1, b5 is Element of bool the carrier of b1: b5 in b2}
   holds b3 is basis of b1;

:: RLTOPSP1:funcnot 4 => RLTOPSP1:func 4
definition
  let a1 be non empty RLTopStruct;
  let a2 be Element of REAL;
  func mlt(A2,A1) -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 means
    for b1 being Element of the carrier of a1 holds
       it . b1 = a2 * b1;
end;

:: RLTOPSP1:def 13
theorem
for b1 being non empty RLTopStruct
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
      b3 = mlt(b2,b1)
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = b2 * b4;

:: RLTOPSP1:th 47
theorem
for b1 being non empty RLTopStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty Element of REAL holds
   (mlt(b3,b1)) .: b2 = b3 * b2;

:: RLTOPSP1:th 48
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being non empty Element of REAL holds
   rng mlt(b2,b1) = [#] b1;

:: RLTOPSP1:th 49
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being non empty Element of REAL holds
   (mlt(b2,b1)) /" = mlt(b2 ",b1);

:: RLTOPSP1:funcreg 12
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be non empty Element of REAL;
  cluster mlt(a2,a1) -> Function-like quasi_total being_homeomorphism;
end;

:: RLTOPSP1:th 50
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being open Element of bool the carrier of b1
for b3 being non empty Element of REAL holds
   b3 * b2 is open(b1);

:: RLTOPSP1:th 51
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being closed Element of bool the carrier of b1
for b3 being non empty Element of REAL holds
   b3 * b2 is closed(b1);

:: RLTOPSP1:th 52
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty Element of REAL holds
   b3 * Int b2 = Int (b3 * b2);

:: RLTOPSP1:th 53
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty Element of REAL holds
   b3 * Cl b2 = Cl (b3 * b2);

:: RLTOPSP1:th 54
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of bool the carrier of b1
      st b1 is being_T1
   holds 0 * Cl b2 = Cl (0 * b2);

:: RLTOPSP1:th 55
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of the carrier of b1
for b3 being a_neighborhood of b2
for b4 being non empty Element of REAL holds
   b4 * b3 is a_neighborhood of b4 * b2;

:: RLTOPSP1:th 56
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being a_neighborhood of 0. b1
for b3 being non empty Element of REAL holds
   b3 * b2 is a_neighborhood of 0. b1;

:: RLTOPSP1:funcreg 13
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be bounded Element of bool the carrier of a1;
  let a3 be Element of REAL;
  cluster a3 * a2 -> bounded;
end;

:: RLTOPSP1:th 57
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being a_neighborhood of 0. b1 holds
   ex b3 being open a_neighborhood of 0. b1 st
      b3 is symmetric(b1) & b3 + b3 c= b2;

:: RLTOPSP1:th 58
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being compact Element of bool the carrier of b1
for b3 being closed Element of bool the carrier of b1
      st b2 misses b3
   holds ex b4 being a_neighborhood of 0. b1 st
      b2 + b4 misses b3 + b4;

:: RLTOPSP1:th 59
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being basis of 0. b1
for b3 being a_neighborhood of 0. b1 holds
   ex b4 being a_neighborhood of 0. b1 st
      b4 in b2 & Cl b4 c= b3;

:: RLTOPSP1:th 60
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being a_neighborhood of 0. b1 holds
   ex b3 being a_neighborhood of 0. b1 st
      Cl b3 c= b2;

:: RLTOPSP1:condreg 2
registration
  cluster non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like being_T1 add-continuous Mult-continuous -> being_T2 (RLTopStruct);
end;

:: RLTOPSP1:th 61
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being Element of bool the carrier of b1 holds
   Cl b2 = meet {b2 + b3 where b3 is a_neighborhood of 0. b1: TRUE};

:: RLTOPSP1:th 62
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Int b2) + Int b3 c= Int (b2 + b3);

:: RLTOPSP1:th 63
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2, b3 being Element of bool the carrier of b1 holds
(Cl b2) + Cl b3 c= Cl (b2 + b3);

:: RLTOPSP1:funcreg 14
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be convex Element of bool the carrier of a1;
  cluster Cl a2 -> convex;
end;

:: RLTOPSP1:funcreg 15
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be convex Element of bool the carrier of a1;
  cluster Int a2 -> convex;
end;

:: RLTOPSP1:funcreg 16
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be circled Element of bool the carrier of a1;
  cluster Cl a2 -> circled;
end;

:: RLTOPSP1:th 64
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being circled Element of bool the carrier of b1
      st 0. b1 in Int b2
   holds Int b2 is circled(b1);

:: RLTOPSP1:funcreg 17
registration
  let a1 be non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct;
  let a2 be bounded Element of bool the carrier of a1;
  cluster Cl a2 -> bounded;
end;

:: RLTOPSP1:th 65
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being a_neighborhood of 0. b1 holds
   ex b3 being a_neighborhood of 0. b1 st
      b3 is circled(b1) & b3 c= b2;

:: RLTOPSP1:th 66
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
for b2 being a_neighborhood of 0. b1
      st b2 is convex(b1)
   holds ex b3 being a_neighborhood of 0. b1 st
      b3 is circled(b1) & b3 is convex(b1) & b3 c= b2;

:: RLTOPSP1:th 67
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct holds
   ex b2 being basis of 0. b1 st
      b2 is circled-membered(b1);

:: RLTOPSP1:th 68
theorem
for b1 being non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like add-continuous Mult-continuous RLTopStruct
      st b1 is locally-convex
   holds ex b2 being basis of 0. b1 st
      b2 is convex-membered(b1);