Article URYSOHN3, MML version 4.99.1005

:: URYSOHN3:th 1
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
   st b2 <> {} & b2 misses b3
for b4 being Element of NAT holds
   ex b5 being Function-like quasi_total Relation of dyadic b4,bool the carrier of b1 st
      b2 c= b5 . 0 &
       b3 = ([#] b1) \ (b5 . 1) &
       (for b6, b7 being Element of dyadic b4
             st b6 < b7
          holds b5 . b6 is open(b1) & b5 . b7 is open(b1) & Cl (b5 . b6) c= b5 . b7);

:: URYSOHN3:modenot 1 => URYSOHN3:mode 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  let a4 be Element of NAT;
  assume a1 is being_T4 & a2 <> {} & a2 is closed(a1) & a3 is closed(a1) & a2 misses a3;
  mode Drizzle of A2,A3,A4 -> Function-like quasi_total Relation of dyadic a4,bool the carrier of a1 means
    a2 c= it . 0 &
     a3 = ([#] a1) \ (it . 1) &
     (for b1, b2 being Element of dyadic a4
           st b1 < b2
        holds it . b1 is open(a1) & it . b2 is open(a1) & Cl (it . b1) c= it . b2);
end;

:: URYSOHN3:dfs 1
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  let a4 be Element of NAT;
  let a5 be Function-like quasi_total Relation of dyadic a4,bool the carrier of a1;
To prove
     a5 is Drizzle of a2,a3,a4
it is sufficient to prove
thus a1 is being_T4 & a2 <> {} & a2 is closed(a1) & a3 is closed(a1) & a2 misses a3;
  thus a2 c= a5 . 0 &
     a3 = ([#] a1) \ (a5 . 1) &
     (for b1, b2 being Element of dyadic a4
           st b1 < b2
        holds a5 . b1 is open(a1) & a5 . b2 is open(a1) & Cl (a5 . b1) c= a5 . b2);

:: URYSOHN3:def 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of NAT
   st b1 is being_T4 & b2 <> {} & b2 is closed(b1) & b3 is closed(b1) & b2 misses b3
for b5 being Function-like quasi_total Relation of dyadic b4,bool the carrier of b1 holds
      b5 is Drizzle of b2,b3,b4
   iff
      b2 c= b5 . 0 &
       b3 = ([#] b1) \ (b5 . 1) &
       (for b6, b7 being Element of dyadic b4
             st b6 < b7
          holds b5 . b6 is open(b1) & b5 . b7 is open(b1) & Cl (b5 . b6) c= b5 . b7);

:: URYSOHN3:th 3
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
   st b2 <> {} & b2 misses b3
for b4 being Element of NAT
for b5 being Drizzle of b2,b3,b4 holds
   ex b6 being Drizzle of b2,b3,b4 + 1 st
      for b7 being Element of dyadic (b4 + 1)
            st b7 in dyadic b4
         holds b6 . b7 = b5 . b7;

:: URYSOHN3:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of NAT
for b5 being Drizzle of b2,b3,b4 holds
   b5 is Element of PFuncs(DYADIC,bool the carrier of b1);

:: URYSOHN3:funcnot 1 => URYSOHN3:func 1
definition
  let a1, a2 be non empty set;
  let a3 be Function-like quasi_total Relation of NAT,PFuncs(a1,a2);
  let a4 be Element of NAT;
  redefine func a3 . a4 -> Element of PFuncs(a1,a2);
end;

:: URYSOHN3:th 5
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
      st b2 <> {} & b2 misses b3
   holds ex b4 being Functional_Sequence of DYADIC,bool the carrier of b1 st
      for b5 being Element of NAT holds
         b4 . b5 is Drizzle of b2,b3,b5 &
          (for b6 being Element of proj1 (b4 . b5) holds
             (b4 . b5) . b6 = (b4 . (b5 + 1)) . b6);

:: URYSOHN3:modenot 2 => URYSOHN3:mode 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  assume a1 is being_T4 & a2 <> {} & a2 is closed(a1) & a3 is closed(a1) & a2 misses a3;
  mode Rain of A2,A3 -> Functional_Sequence of DYADIC,bool the carrier of a1 means
    for b1 being Element of NAT holds
       it . b1 is Drizzle of a2,a3,b1 &
        (for b2 being Element of proj1 (it . b1) holds
           (it . b1) . b2 = (it . (b1 + 1)) . b2);
end;

:: URYSOHN3:dfs 2
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  let a4 be Functional_Sequence of DYADIC,bool the carrier of a1;
To prove
     a4 is Rain of a2,a3
it is sufficient to prove
thus a1 is being_T4 & a2 <> {} & a2 is closed(a1) & a3 is closed(a1) & a2 misses a3;
  thus for b1 being Element of NAT holds
       a4 . b1 is Drizzle of a2,a3,b1 &
        (for b2 being Element of proj1 (a4 . b1) holds
           (a4 . b1) . b2 = (a4 . (b1 + 1)) . b2);

:: URYSOHN3:def 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
   st b1 is being_T4 & b2 <> {} & b2 is closed(b1) & b3 is closed(b1) & b2 misses b3
for b4 being Functional_Sequence of DYADIC,bool the carrier of b1 holds
      b4 is Rain of b2,b3
   iff
      for b5 being Element of NAT holds
         b4 . b5 is Drizzle of b2,b3,b5 &
          (for b6 being Element of proj1 (b4 . b5) holds
             (b4 . b5) . b6 = (b4 . (b5 + 1)) . b6);

:: URYSOHN3:funcnot 2 => URYSOHN3:func 2
definition
  let a1 be Element of REAL;
  assume a1 in DYADIC;
  func inf_number_dyadic A1 -> Element of NAT means
    (a1 in dyadic 0 implies it = 0) &
     (it = 0 implies a1 in dyadic 0) &
     (for b1 being Element of NAT
           st a1 in dyadic (b1 + 1) & not a1 in dyadic b1
        holds it = b1 + 1);
end;

:: URYSOHN3:def 3
theorem
for b1 being Element of REAL
   st b1 in DYADIC
for b2 being Element of NAT holds
      b2 = inf_number_dyadic b1
   iff
      (b1 in dyadic 0 implies b2 = 0) &
       (b2 = 0 implies b1 in dyadic 0) &
       (for b3 being Element of NAT
             st b1 in dyadic (b3 + 1) & not b1 in dyadic b3
          holds b2 = b3 + 1);

:: URYSOHN3:th 6
theorem
for b1 being Element of REAL
      st b1 in DYADIC
   holds b1 in dyadic inf_number_dyadic b1;

:: URYSOHN3:th 7
theorem
for b1 being Element of REAL
   st b1 in DYADIC
for b2 being Element of NAT
      st inf_number_dyadic b1 <= b2
   holds b1 in dyadic b2;

:: URYSOHN3:th 8
theorem
for b1 being Element of REAL
for b2 being Element of NAT
      st b1 in dyadic b2
   holds inf_number_dyadic b1 <= b2;

:: URYSOHN3:th 9
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
   st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of REAL
   st b5 in DYADIC
for b6 being Element of NAT holds
   (b4 . inf_number_dyadic b5) . b5 = (b4 . ((inf_number_dyadic b5) + b6)) . b5;

:: URYSOHN3:th 10
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
   st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of REAL
      st b5 in DYADIC
   holds ex b6 being Element of bool the carrier of b1 st
      for b7 being Element of NAT
            st b5 in dyadic b7
         holds b6 = (b4 . b7) . b5;

:: URYSOHN3:th 11
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
   st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3 holds
   ex b5 being Function-like quasi_total Relation of DOM,bool the carrier of b1 st
      for b6 being Element of REAL
            st b6 in DOM
         holds (b6 in halfline 0 implies b5 . b6 = {}) &
          (b6 in right_open_halfline 1 implies b5 . b6 = the carrier of b1) &
          (b6 in DYADIC implies for b7 being Element of NAT
                st b6 in dyadic b7
             holds b5 . b6 = (b4 . b7) . b6);

:: URYSOHN3:funcnot 3 => URYSOHN3:func 3
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  let a4 be Rain of a2,a3;
  assume a1 is being_T4 & a2 <> {} & a2 is closed(a1) & a3 is closed(a1) & a2 misses a3;
  func Tempest A4 -> Function-like quasi_total Relation of DOM,bool the carrier of a1 means
    for b1 being Element of REAL
          st b1 in DOM
       holds (b1 in halfline 0 implies it . b1 = {}) &
        (b1 in right_open_halfline 1 implies it . b1 = the carrier of a1) &
        (b1 in DYADIC implies for b2 being Element of NAT
              st b1 in dyadic b2
           holds it . b1 = (a4 . b2) . b1);
end;

:: URYSOHN3:def 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
   st b1 is being_T4 & b2 <> {} & b2 is closed(b1) & b3 is closed(b1) & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Function-like quasi_total Relation of DOM,bool the carrier of b1 holds
      b5 = Tempest b4
   iff
      for b6 being Element of REAL
            st b6 in DOM
         holds (b6 in halfline 0 implies b5 . b6 = {}) &
          (b6 in right_open_halfline 1 implies b5 . b6 = the carrier of b1) &
          (b6 in DYADIC implies for b7 being Element of NAT
                st b6 in dyadic b7
             holds b5 . b6 = (b4 . b7) . b6);

:: URYSOHN3:funcnot 4 => URYSOHN3:func 4
definition
  let a1 be non empty set;
  let a2 be TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of a1,bool the carrier of a2;
  let a4 be Element of a1;
  redefine func a3 . a4 -> Element of bool the carrier of a2;
end;

:: URYSOHN3:th 12
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
   st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of REAL
for b6 being Element of bool the carrier of b1
      st b6 = (Tempest b4) . b5 & b5 in DOM
   holds b6 is open(b1);

:: URYSOHN3:th 13
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
   st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5, b6 being Element of REAL
   st b5 in DOM & b6 in DOM & b5 < b6
for b7 being Element of bool the carrier of b1
      st b7 = (Tempest b4) . b5
   holds Cl b7 c= (Tempest b4) . b6;

:: URYSOHN3:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3
for b5 being Element of the carrier of b1 holds
   ex b6 being Element of bool ExtREAL st
      for b7 being set holds
            b7 in b6
         iff
            b7 in DYADIC &
             (for b8 being Element of REAL
                   st b8 = b7
                holds not b5 in (Tempest b4) . b8);

:: URYSOHN3:funcnot 5 => URYSOHN3:func 5
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  let a4 be Rain of a2,a3;
  let a5 be Element of the carrier of a1;
  func Rainbow(A5,A4) -> Element of bool ExtREAL means
    for b1 being set holds
          b1 in it
       iff
          b1 in DYADIC &
           (for b2 being Element of REAL
                 st b2 = b1
              holds not a5 in (Tempest a4) . b2);
end;

:: URYSOHN3:def 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3
for b5 being Element of the carrier of b1
for b6 being Element of bool ExtREAL holds
      b6 = Rainbow(b5,b4)
   iff
      for b7 being set holds
            b7 in b6
         iff
            b7 in DYADIC &
             (for b8 being Element of REAL
                   st b8 = b7
                holds not b5 in (Tempest b4) . b8);

:: URYSOHN3:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3
for b5 being Element of the carrier of b1 holds
   Rainbow(b5,b4) c= DYADIC;

:: URYSOHN3:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3 holds
   ex b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      for b6 being Element of the carrier of b1 holds
         (Rainbow(b6,b4) = {} implies b5 . b6 = 0) &
          (for b7 being non empty Element of bool ExtREAL
                st b7 = Rainbow(b6,b4)
             holds b5 . b6 = sup b7);

:: URYSOHN3:funcnot 6 => URYSOHN3:func 6
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  let a4 be Rain of a2,a3;
  func Thunder A4 -> Function-like quasi_total Relation of the carrier of a1,the carrier of R^1 means
    for b1 being Element of the carrier of a1 holds
       (Rainbow(b1,a4) = {} implies it . b1 = 0) &
        (for b2 being non empty Element of bool ExtREAL
              st b2 = Rainbow(b1,a4)
           holds it . b1 = sup b2);
end;

:: URYSOHN3:def 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 holds
      b5 = Thunder b4
   iff
      for b6 being Element of the carrier of b1 holds
         (Rainbow(b6,b4) = {} implies b5 . b6 = 0) &
          (for b7 being non empty Element of bool ExtREAL
                st b7 = Rainbow(b6,b4)
             holds b5 . b6 = sup b7);

:: URYSOHN3:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Rain of b2,b3
for b5 being Element of the carrier of b1
for b6 being non empty Element of bool ExtREAL
   st b6 = Rainbow(b5,b4)
for b7 being Element of ExtREAL
      st b7 = 1
   holds 0. <= sup b6 & sup b6 <= b7;

:: URYSOHN3:th 18
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
   st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of DOM
for b6 being Element of the carrier of b1
      st (Thunder b4) . b6 < b5
   holds b6 in (Tempest b4) . b5;

:: URYSOHN3:th 19
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
   st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of REAL
   st b5 in DYADIC \/ right_open_halfline 1 & 0 < b5
for b6 being Element of the carrier of b1
      st b6 in (Tempest b4) . b5
   holds (Thunder b4) . b6 <= b5;

:: URYSOHN3:th 20
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
   st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3
for b5 being Element of DOM
   st 0 < b5
for b6 being Element of the carrier of b1
      st b5 < (Thunder b4) . b6
   holds not b6 in (Tempest b4) . b5;

:: URYSOHN3:th 21
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
   st b2 <> {} & b2 misses b3
for b4 being Rain of b2,b3 holds
   Thunder b4 is continuous(b1, R^1) &
    (for b5 being Element of the carrier of b1 holds
       0 <= (Thunder b4) . b5 & (Thunder b4) . b5 <= 1 & (b5 in b2 implies (Thunder b4) . b5 = 0) & (b5 in b3 implies (Thunder b4) . b5 = 1));

:: URYSOHN3:th 22
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
      st b2 <> {} & b2 misses b3
   holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      b4 is continuous(b1, R^1) &
       (for b5 being Element of the carrier of b1 holds
          0 <= b4 . b5 & b4 . b5 <= 1 & (b5 in b2 implies b4 . b5 = 0) & (b5 in b3 implies b4 . b5 = 1));

:: URYSOHN3:th 23
theorem
for b1 being non empty TopSpace-like being_T4 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
      st b2 misses b3
   holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      b4 is continuous(b1, R^1) &
       (for b5 being Element of the carrier of b1 holds
          0 <= b4 . b5 & b4 . b5 <= 1 & (b5 in b2 implies b4 . b5 = 0) & (b5 in b3 implies b4 . b5 = 1));

:: URYSOHN3:th 24
theorem
for b1 being non empty TopSpace-like compact being_T2 TopStruct
for b2, b3 being closed Element of bool the carrier of b1
      st b2 misses b3
   holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
      b4 is continuous(b1, R^1) &
       (for b5 being Element of the carrier of b1 holds
          0 <= b4 . b5 & b4 . b5 <= 1 & (b5 in b2 implies b4 . b5 = 0) & (b5 in b3 implies b4 . b5 = 1));