Article DECOMP_1, MML version 4.99.1005

:: DECOMP_1:modenot 1 => DECOMP_1:mode 1
definition
  let a1 be TopStruct;
  mode alpha-set of A1 -> Element of bool the carrier of a1 means
    it c= Int Cl Int it;
end;

:: DECOMP_1:dfs 1
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is alpha-set of a1
it is sufficient to prove
  thus a2 c= Int Cl Int a2;

:: DECOMP_1:def 1
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is alpha-set of b1
   iff
      b2 c= Int Cl Int b2;

:: DECOMP_1:attrnot 1 => DECOMP_1:attr 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is semi-open means
    a2 c= Cl Int a2;
end;

:: DECOMP_1:dfs 2
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is semi-open
it is sufficient to prove
  thus a2 c= Cl Int a2;

:: DECOMP_1:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is semi-open(b1)
   iff
      b2 c= Cl Int b2;

:: DECOMP_1:attrnot 2 => DECOMP_1:attr 2
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is pre-open means
    a2 c= Int Cl a2;
end;

:: DECOMP_1:dfs 3
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is pre-open
it is sufficient to prove
  thus a2 c= Int Cl a2;

:: DECOMP_1:def 3
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is pre-open(b1)
   iff
      b2 c= Int Cl b2;

:: DECOMP_1:attrnot 3 => DECOMP_1:attr 3
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is pre-semi-open means
    a2 c= Cl Int Cl a2;
end;

:: DECOMP_1:dfs 4
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is pre-semi-open
it is sufficient to prove
  thus a2 c= Cl Int Cl a2;

:: DECOMP_1:def 4
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is pre-semi-open(b1)
   iff
      b2 c= Cl Int Cl b2;

:: DECOMP_1:attrnot 4 => DECOMP_1:attr 4
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is semi-pre-open means
    a2 c= (Cl Int a2) \/ Int Cl a2;
end;

:: DECOMP_1:dfs 5
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is semi-pre-open
it is sufficient to prove
  thus a2 c= (Cl Int a2) \/ Int Cl a2;

:: DECOMP_1:def 5
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is semi-pre-open(b1)
   iff
      b2 c= (Cl Int b2) \/ Int Cl b2;

:: DECOMP_1:funcnot 1 => DECOMP_1:func 1
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  func sInt A2 -> Element of bool the carrier of a1 equals
    a2 /\ Cl Int a2;
end;

:: DECOMP_1:def 6
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   sInt b2 = b2 /\ Cl Int b2;

:: DECOMP_1:funcnot 2 => DECOMP_1:func 2
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  func pInt A2 -> Element of bool the carrier of a1 equals
    a2 /\ Int Cl a2;
end;

:: DECOMP_1:def 7
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   pInt b2 = b2 /\ Int Cl b2;

:: DECOMP_1:funcnot 3 => DECOMP_1:func 3
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  func alphaInt A2 -> Element of bool the carrier of a1 equals
    a2 /\ Int Cl Int a2;
end;

:: DECOMP_1:def 8
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   alphaInt b2 = b2 /\ Int Cl Int b2;

:: DECOMP_1:funcnot 4 => DECOMP_1:func 4
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  func psInt A2 -> Element of bool the carrier of a1 equals
    a2 /\ Cl Int Cl a2;
end;

:: DECOMP_1:def 9
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   psInt b2 = b2 /\ Cl Int Cl b2;

:: DECOMP_1:funcnot 5 => DECOMP_1:func 5
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  func spInt A2 -> Element of bool the carrier of a1 equals
    (sInt a2) \/ pInt a2;
end;

:: DECOMP_1:def 10
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
   spInt b2 = (sInt b2) \/ pInt b2;

:: DECOMP_1:funcnot 6 => DECOMP_1:func 6
definition
  let a1 be TopSpace-like TopStruct;
  func A1 ^alpha -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: b1 is alpha-set of a1};
end;

:: DECOMP_1:def 11
theorem
for b1 being TopSpace-like TopStruct holds
   b1 ^alpha = {b2 where b2 is Element of bool the carrier of b1: b2 is alpha-set of b1};

:: DECOMP_1:funcnot 7 => DECOMP_1:func 7
definition
  let a1 be TopSpace-like TopStruct;
  func SO A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: b1 is semi-open(a1)};
end;

:: DECOMP_1:def 12
theorem
for b1 being TopSpace-like TopStruct holds
   SO b1 = {b2 where b2 is Element of bool the carrier of b1: b2 is semi-open(b1)};

:: DECOMP_1:funcnot 8 => DECOMP_1:func 8
definition
  let a1 be TopSpace-like TopStruct;
  func PO A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: b1 is pre-open(a1)};
end;

:: DECOMP_1:def 13
theorem
for b1 being TopSpace-like TopStruct holds
   PO b1 = {b2 where b2 is Element of bool the carrier of b1: b2 is pre-open(b1)};

:: DECOMP_1:funcnot 9 => DECOMP_1:func 9
definition
  let a1 be TopSpace-like TopStruct;
  func SPO A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: b1 is semi-pre-open(a1)};
end;

:: DECOMP_1:def 14
theorem
for b1 being TopSpace-like TopStruct holds
   SPO b1 = {b2 where b2 is Element of bool the carrier of b1: b2 is semi-pre-open(b1)};

:: DECOMP_1:funcnot 10 => DECOMP_1:func 10
definition
  let a1 be TopSpace-like TopStruct;
  func PSO A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: b1 is pre-semi-open(a1)};
end;

:: DECOMP_1:def 15
theorem
for b1 being TopSpace-like TopStruct holds
   PSO b1 = {b2 where b2 is Element of bool the carrier of b1: b2 is pre-semi-open(b1)};

:: DECOMP_1:funcnot 11 => DECOMP_1:func 11
definition
  let a1 be TopSpace-like TopStruct;
  func D(c,alpha) A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: Int b1 = alphaInt b1};
end;

:: DECOMP_1:def 16
theorem
for b1 being TopSpace-like TopStruct holds
   D(c,alpha) b1 = {b2 where b2 is Element of bool the carrier of b1: Int b2 = alphaInt b2};

:: DECOMP_1:funcnot 12 => DECOMP_1:func 12
definition
  let a1 be TopSpace-like TopStruct;
  func D(c,p) A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: Int b1 = pInt b1};
end;

:: DECOMP_1:def 17
theorem
for b1 being TopSpace-like TopStruct holds
   D(c,p) b1 = {b2 where b2 is Element of bool the carrier of b1: Int b2 = pInt b2};

:: DECOMP_1:funcnot 13 => DECOMP_1:func 13
definition
  let a1 be TopSpace-like TopStruct;
  func D(c,s) A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: Int b1 = sInt b1};
end;

:: DECOMP_1:def 18
theorem
for b1 being TopSpace-like TopStruct holds
   D(c,s) b1 = {b2 where b2 is Element of bool the carrier of b1: Int b2 = sInt b2};

:: DECOMP_1:funcnot 14 => DECOMP_1:func 14
definition
  let a1 be TopSpace-like TopStruct;
  func D(c,ps) A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: Int b1 = psInt b1};
end;

:: DECOMP_1:def 19
theorem
for b1 being TopSpace-like TopStruct holds
   D(c,ps) b1 = {b2 where b2 is Element of bool the carrier of b1: Int b2 = psInt b2};

:: DECOMP_1:funcnot 15 => DECOMP_1:func 15
definition
  let a1 be TopSpace-like TopStruct;
  func D(alpha,p) A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: alphaInt b1 = pInt b1};
end;

:: DECOMP_1:def 20
theorem
for b1 being TopSpace-like TopStruct holds
   D(alpha,p) b1 = {b2 where b2 is Element of bool the carrier of b1: alphaInt b2 = pInt b2};

:: DECOMP_1:funcnot 16 => DECOMP_1:func 16
definition
  let a1 be TopSpace-like TopStruct;
  func D(alpha,s) A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: alphaInt b1 = sInt b1};
end;

:: DECOMP_1:def 21
theorem
for b1 being TopSpace-like TopStruct holds
   D(alpha,s) b1 = {b2 where b2 is Element of bool the carrier of b1: alphaInt b2 = sInt b2};

:: DECOMP_1:funcnot 17 => DECOMP_1:func 17
definition
  let a1 be TopSpace-like TopStruct;
  func D(alpha,ps) A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: alphaInt b1 = psInt b1};
end;

:: DECOMP_1:def 22
theorem
for b1 being TopSpace-like TopStruct holds
   D(alpha,ps) b1 = {b2 where b2 is Element of bool the carrier of b1: alphaInt b2 = psInt b2};

:: DECOMP_1:funcnot 18 => DECOMP_1:func 18
definition
  let a1 be TopSpace-like TopStruct;
  func D(p,sp) A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: pInt b1 = spInt b1};
end;

:: DECOMP_1:def 23
theorem
for b1 being TopSpace-like TopStruct holds
   D(p,sp) b1 = {b2 where b2 is Element of bool the carrier of b1: pInt b2 = spInt b2};

:: DECOMP_1:funcnot 19 => DECOMP_1:func 19
definition
  let a1 be TopSpace-like TopStruct;
  func D(p,ps) A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: pInt b1 = psInt b1};
end;

:: DECOMP_1:def 24
theorem
for b1 being TopSpace-like TopStruct holds
   D(p,ps) b1 = {b2 where b2 is Element of bool the carrier of b1: pInt b2 = psInt b2};

:: DECOMP_1:funcnot 20 => DECOMP_1:func 20
definition
  let a1 be TopSpace-like TopStruct;
  func D(sp,ps) A1 -> Element of bool bool the carrier of a1 equals
    {b1 where b1 is Element of bool the carrier of a1: spInt b1 = psInt b1};
end;

:: DECOMP_1:def 25
theorem
for b1 being TopSpace-like TopStruct holds
   D(sp,ps) b1 = {b2 where b2 is Element of bool the carrier of b1: spInt b2 = psInt b2};

:: DECOMP_1:th 1
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      alphaInt b2 = pInt b2
   iff
      sInt b2 = psInt b2;

:: DECOMP_1:th 2
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is alpha-set of b1
   iff
      b2 = alphaInt b2;

:: DECOMP_1:th 3
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is semi-open(b1)
   iff
      b2 = sInt b2;

:: DECOMP_1:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is pre-open(b1)
   iff
      b2 = pInt b2;

:: DECOMP_1:th 5
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is pre-semi-open(b1)
   iff
      b2 = psInt b2;

:: DECOMP_1:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is semi-pre-open(b1)
   iff
      b2 = spInt b2;

:: DECOMP_1:th 7
theorem
for b1 being TopSpace-like TopStruct holds
   b1 ^alpha /\ D(c,alpha) b1 = the topology of b1;

:: DECOMP_1:th 8
theorem
for b1 being TopSpace-like TopStruct holds
   (SO b1) /\ D(c,s) b1 = the topology of b1;

:: DECOMP_1:th 9
theorem
for b1 being TopSpace-like TopStruct holds
   (PO b1) /\ D(c,p) b1 = the topology of b1;

:: DECOMP_1:th 10
theorem
for b1 being TopSpace-like TopStruct holds
   (PSO b1) /\ D(c,ps) b1 = the topology of b1;

:: DECOMP_1:th 11
theorem
for b1 being TopSpace-like TopStruct holds
   (PO b1) /\ D(alpha,p) b1 = b1 ^alpha;

:: DECOMP_1:th 12
theorem
for b1 being TopSpace-like TopStruct holds
   (SO b1) /\ D(alpha,s) b1 = b1 ^alpha;

:: DECOMP_1:th 13
theorem
for b1 being TopSpace-like TopStruct holds
   (PSO b1) /\ D(alpha,ps) b1 = b1 ^alpha;

:: DECOMP_1:th 14
theorem
for b1 being TopSpace-like TopStruct holds
   (SPO b1) /\ D(p,sp) b1 = PO b1;

:: DECOMP_1:th 15
theorem
for b1 being TopSpace-like TopStruct holds
   (PSO b1) /\ D(p,ps) b1 = PO b1;

:: DECOMP_1:th 16
theorem
for b1 being TopSpace-like TopStruct holds
   (PSO b1) /\ D(alpha,p) b1 = SO b1;

:: DECOMP_1:th 17
theorem
for b1 being TopSpace-like TopStruct holds
   (PSO b1) /\ D(sp,ps) b1 = SPO b1;

:: DECOMP_1:attrnot 5 => DECOMP_1:attr 5
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is s-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in SO a1;
end;

:: DECOMP_1:dfs 26
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is s-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in SO a1;

:: DECOMP_1:def 26
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is s-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in SO b1;

:: DECOMP_1:attrnot 6 => DECOMP_1:attr 6
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is p-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in PO a1;
end;

:: DECOMP_1:dfs 27
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is p-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in PO a1;

:: DECOMP_1:def 27
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is p-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in PO b1;

:: DECOMP_1:attrnot 7 => DECOMP_1:attr 7
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is alpha-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in a1 ^alpha;
end;

:: DECOMP_1:dfs 28
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is alpha-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in a1 ^alpha;

:: DECOMP_1:def 28
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is alpha-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in b1 ^alpha;

:: DECOMP_1:attrnot 8 => DECOMP_1:attr 8
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is ps-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in PSO a1;
end;

:: DECOMP_1:dfs 29
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is ps-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in PSO a1;

:: DECOMP_1:def 29
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is ps-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in PSO b1;

:: DECOMP_1:attrnot 9 => DECOMP_1:attr 9
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is sp-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in SPO a1;
end;

:: DECOMP_1:dfs 30
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is sp-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in SPO a1;

:: DECOMP_1:def 30
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is sp-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in SPO b1;

:: DECOMP_1:attrnot 10 => DECOMP_1:attr 10
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is (c,alpha)-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(c,alpha) a1;
end;

:: DECOMP_1:dfs 31
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is (c,alpha)-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(c,alpha) a1;

:: DECOMP_1:def 31
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is (c,alpha)-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in D(c,alpha) b1;

:: DECOMP_1:attrnot 11 => DECOMP_1:attr 11
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is (c,s)-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(c,s) a1;
end;

:: DECOMP_1:dfs 32
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is (c,s)-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(c,s) a1;

:: DECOMP_1:def 32
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is (c,s)-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in D(c,s) b1;

:: DECOMP_1:attrnot 12 => DECOMP_1:attr 12
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is (c,p)-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(c,p) a1;
end;

:: DECOMP_1:dfs 33
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is (c,p)-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(c,p) a1;

:: DECOMP_1:def 33
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is (c,p)-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in D(c,p) b1;

:: DECOMP_1:attrnot 13 => DECOMP_1:attr 13
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is (c,ps)-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(c,ps) a1;
end;

:: DECOMP_1:dfs 34
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is (c,ps)-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(c,ps) a1;

:: DECOMP_1:def 34
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is (c,ps)-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in D(c,ps) b1;

:: DECOMP_1:attrnot 14 => DECOMP_1:attr 14
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is (alpha,p)-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(alpha,p) a1;
end;

:: DECOMP_1:dfs 35
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is (alpha,p)-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(alpha,p) a1;

:: DECOMP_1:def 35
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is (alpha,p)-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in D(alpha,p) b1;

:: DECOMP_1:attrnot 15 => DECOMP_1:attr 15
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is (alpha,s)-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(alpha,s) a1;
end;

:: DECOMP_1:dfs 36
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is (alpha,s)-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(alpha,s) a1;

:: DECOMP_1:def 36
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is (alpha,s)-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in D(alpha,s) b1;

:: DECOMP_1:attrnot 16 => DECOMP_1:attr 16
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is (alpha,ps)-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(alpha,ps) a1;
end;

:: DECOMP_1:dfs 37
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is (alpha,ps)-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(alpha,ps) a1;

:: DECOMP_1:def 37
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is (alpha,ps)-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in D(alpha,ps) b1;

:: DECOMP_1:attrnot 17 => DECOMP_1:attr 17
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is (p,ps)-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(p,ps) a1;
end;

:: DECOMP_1:dfs 38
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is (p,ps)-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(p,ps) a1;

:: DECOMP_1:def 38
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is (p,ps)-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in D(p,ps) b1;

:: DECOMP_1:attrnot 18 => DECOMP_1:attr 18
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is (p,sp)-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(p,sp) a1;
end;

:: DECOMP_1:dfs 39
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is (p,sp)-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(p,sp) a1;

:: DECOMP_1:def 39
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is (p,sp)-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in D(p,sp) b1;

:: DECOMP_1:attrnot 19 => DECOMP_1:attr 19
definition
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is (sp,ps)-continuous means
    for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(sp,ps) a1;
end;

:: DECOMP_1:dfs 40
definiens
  let a1, a2 be non empty TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is (sp,ps)-continuous
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a2
          st b1 is open(a2)
       holds a3 " b1 in D(sp,ps) a1;

:: DECOMP_1:def 40
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is (sp,ps)-continuous(b1, b2)
   iff
      for b4 being Element of bool the carrier of b2
            st b4 is open(b2)
         holds b3 " b4 in D(sp,ps) b1;

:: DECOMP_1:th 18
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is alpha-continuous(b1, b2)
   iff
      b3 is p-continuous(b1, b2) & b3 is (alpha,p)-continuous(b1, b2);

:: DECOMP_1:th 19
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is alpha-continuous(b1, b2)
   iff
      b3 is s-continuous(b1, b2) & b3 is (alpha,s)-continuous(b1, b2);

:: DECOMP_1:th 20
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is alpha-continuous(b1, b2)
   iff
      b3 is ps-continuous(b1, b2) & b3 is (alpha,ps)-continuous(b1, b2);

:: DECOMP_1:th 21
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is p-continuous(b1, b2)
   iff
      b3 is sp-continuous(b1, b2) & b3 is (p,sp)-continuous(b1, b2);

:: DECOMP_1:th 22
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is p-continuous(b1, b2)
   iff
      b3 is ps-continuous(b1, b2) & b3 is (p,ps)-continuous(b1, b2);

:: DECOMP_1:th 23
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is s-continuous(b1, b2)
   iff
      b3 is ps-continuous(b1, b2) & b3 is (alpha,p)-continuous(b1, b2);

:: DECOMP_1:th 24
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is sp-continuous(b1, b2)
   iff
      b3 is ps-continuous(b1, b2) & b3 is (sp,ps)-continuous(b1, b2);

:: DECOMP_1:th 25
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is continuous(b1, b2)
   iff
      b3 is alpha-continuous(b1, b2) & b3 is (c,alpha)-continuous(b1, b2);

:: DECOMP_1:th 26
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is continuous(b1, b2)
   iff
      b3 is s-continuous(b1, b2) & b3 is (c,s)-continuous(b1, b2);

:: DECOMP_1:th 27
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is continuous(b1, b2)
   iff
      b3 is p-continuous(b1, b2) & b3 is (c,p)-continuous(b1, b2);

:: DECOMP_1:th 28
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is continuous(b1, b2)
   iff
      b3 is ps-continuous(b1, b2) & b3 is (c,ps)-continuous(b1, b2);