Article WAYBEL17, MML version 4.99.1005

:: WAYBEL17:funcnot 1 => WAYBEL17:func 1
definition
  let a1 be non empty set;
  let a2, a3 be Element of a1;
  func (A2,A3),... -> Function-like quasi_total Relation of NAT,a1 means
    for b1 being Element of NAT holds
       (for b2 being Element of NAT holds
           b1 <> 2 * b2 or it . b1 = a2) &
        (for b2 being Element of NAT holds
           b1 <> 2 * b2 implies it . b1 = a3);
end;

:: WAYBEL17:def 1
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being Function-like quasi_total Relation of NAT,b1 holds
      b4 = (b2,b3),...
   iff
      for b5 being Element of NAT holds
         (for b6 being Element of NAT holds
             b5 <> 2 * b6 or b4 . b5 = b2) &
          (for b6 being Element of NAT holds
             b5 <> 2 * b6 implies b4 . b5 = b3);

:: WAYBEL17:sch 1
scheme WAYBEL17:sch 1
{F1 -> non empty TopRelStr,
  F2 -> non empty TopRelStr,
  F3 -> Element of the carrier of F2(),
  F4 -> Relation-like Function-like set}:
F4() .: {F3(b1) where b1 is Element of the carrier of F1(): P1[b1]} = {F4() . F3(b1) where b1 is Element of the carrier of F1(): P1[b1]}
provided
   the carrier of F2() c= proj1 F4();


:: WAYBEL17:sch 2
scheme WAYBEL17:sch 2
{F1 -> reflexive transitive antisymmetric with_suprema with_infima RelStr,
  F2 -> reflexive transitive antisymmetric with_suprema with_infima RelStr,
  F3 -> Element of the carrier of F2(),
  F4 -> Relation-like Function-like set}:
F4() .: {F3(b1) where b1 is Element of the carrier of F1(): P1[b1]} = {F4() . F3(b1) where b1 is Element of the carrier of F1(): P1[b1]}
provided
   the carrier of F2() c= proj1 F4();


:: WAYBEL17:th 1
theorem
for b1, b2 being non empty reflexive RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being lower Element of bool the carrier of b2
      st b3 is monotone(b1, b2)
   holds b3 " b4 is lower(b1);

:: WAYBEL17:th 2
theorem
for b1, b2 being non empty reflexive RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being upper Element of bool the carrier of b2
      st b3 is monotone(b1, b2)
   holds b3 " b4 is upper(b1);

:: WAYBEL17:th 3
theorem
for b1, b2 being non empty reflexive antisymmetric RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is directed-sups-preserving(b1, b2)
   holds b3 is monotone(b1, b2);

:: WAYBEL17:condreg 1
registration
  let a1, a2 be non empty reflexive antisymmetric RelStr;
  cluster Function-like quasi_total directed-sups-preserving -> monotone (Relation of the carrier of a1,the carrier of a2);
end;

:: WAYBEL17:th 4
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima up-complete Scott TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is continuous(b1, b2)
   holds b3 is monotone(b1, b2);

:: WAYBEL17:condreg 2
registration
  let a1, a2 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima up-complete Scott TopRelStr;
  cluster Function-like quasi_total continuous -> monotone (Relation of the carrier of a1,the carrier of a2);
end;

:: WAYBEL17:funcreg 1
registration
  let a1 be set;
  let a2 be reflexive RelStr;
  cluster a1 --> a2 -> reflexive-yielding;
end;

:: WAYBEL17:funcreg 2
registration
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster a2 |^ a1 -> strict complete;
end;

:: WAYBEL17:funcnot 2 => WAYBEL17:func 2
definition
  let a1, a2 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima up-complete Scott TopRelStr;
  func SCMaps(A1,A2) -> strict full SubRelStr of MonMaps(a1,a2) means
    for b1 being Function-like quasi_total Relation of the carrier of a1,the carrier of a2 holds
          b1 in the carrier of it
       iff
          b1 is continuous(a1, a2);
end;

:: WAYBEL17:def 2
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima up-complete Scott TopRelStr
for b3 being strict full SubRelStr of MonMaps(b1,b2) holds
      b3 = SCMaps(b1,b2)
   iff
      for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
            b4 in the carrier of b3
         iff
            b4 is continuous(b1, b2);

:: WAYBEL17:funcreg 3
registration
  let a1, a2 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima up-complete Scott TopRelStr;
  cluster SCMaps(a1,a2) -> non empty strict full;
end;

:: WAYBEL17:funcnot 3 => WAYBEL17:func 3
definition
  let a1 be non empty RelStr;
  let a2, a3 be Element of the carrier of a1;
  func Net-Str(A2,A3) -> non empty strict NetStr over a1 means
    the carrier of it = NAT &
     the mapping of it = (a2,a3),... &
     (for b1, b2 being Element of the carrier of it
     for b3, b4 being Element of NAT
           st b1 = b3 & b2 = b4
        holds    b1 <= b2
        iff
           b3 <= b4);
end;

:: WAYBEL17:def 3
theorem
for b1 being non empty RelStr
for b2, b3 being Element of the carrier of b1
for b4 being non empty strict NetStr over b1 holds
      b4 = Net-Str(b2,b3)
   iff
      the carrier of b4 = NAT &
       the mapping of b4 = (b2,b3),... &
       (for b5, b6 being Element of the carrier of b4
       for b7, b8 being Element of NAT
             st b5 = b7 & b6 = b8
          holds    b5 <= b6
          iff
             b7 <= b8);

:: WAYBEL17:funcreg 4
registration
  let a1 be non empty RelStr;
  let a2, a3 be Element of the carrier of a1;
  cluster Net-Str(a2,a3) -> non empty reflexive transitive antisymmetric strict directed;
end;

:: WAYBEL17:th 5
theorem
for b1 being non empty RelStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of the carrier of Net-Str(b2,b3)
      st (Net-Str(b2,b3)) . b4 <> b2
   holds (Net-Str(b2,b3)) . b4 = b3;

:: WAYBEL17:th 6
theorem
for b1 being non empty RelStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of the carrier of Net-Str(b2,b3)
for b6, b7 being Element of NAT
      st b6 = b4 & b7 = b6 + 1 & b7 = b5
   holds ((Net-Str(b2,b3)) . b4 = b2 implies (Net-Str(b2,b3)) . b5 = b3) &
    ((Net-Str(b2,b3)) . b4 = b3 implies (Net-Str(b2,b3)) . b5 = b2);

:: WAYBEL17:th 7
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2, b3 being Element of the carrier of b1 holds
lim_inf Net-Str(b2,b3) = b2 "/\" b3;

:: WAYBEL17:th 8
theorem
for b1, b2 being reflexive transitive antisymmetric with_infima RelStr
for b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
   lim_inf (b5 * Net-Str(b3,b4)) = (b5 . b3) "/\" (b5 . b4);

:: WAYBEL17:funcnot 4 => WAYBEL17:func 4
definition
  let a1 be non empty RelStr;
  let a2 be non empty Element of bool the carrier of a1;
  func Net-Str A2 -> strict NetStr over a1 equals
    NetStr(#a2,(the InternalRel of a1) |_2 a2,(id the carrier of a1) | a2#);
end;

:: WAYBEL17:def 4
theorem
for b1 being non empty RelStr
for b2 being non empty Element of bool the carrier of b1 holds
   Net-Str b2 = NetStr(#b2,(the InternalRel of b1) |_2 b2,(id the carrier of b1) | b2#);

:: WAYBEL17:th 9
theorem
for b1 being non empty reflexive RelStr
for b2 being non empty Element of bool the carrier of b1 holds
   Net-Str b2 = Net-Str(b2,(id the carrier of b1) | b2);

:: WAYBEL17:funcreg 5
registration
  let a1 be non empty reflexive RelStr;
  let a2 be non empty directed Element of bool the carrier of a1;
  cluster Net-Str a2 -> non empty reflexive strict directed;
end;

:: WAYBEL17:funcreg 6
registration
  let a1 be non empty reflexive transitive RelStr;
  let a2 be non empty directed Element of bool the carrier of a1;
  cluster Net-Str a2 -> transitive strict;
end;

:: WAYBEL17:funcreg 7
registration
  let a1 be non empty reflexive RelStr;
  let a2 be non empty directed Element of bool the carrier of a1;
  cluster Net-Str a2 -> strict monotone;
end;

:: WAYBEL17:th 10
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima up-complete RelStr
for b2 being non empty directed Element of bool the carrier of b1 holds
   lim_inf Net-Str b2 = "\/"(b2,b1);

:: WAYBEL17:th 11
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st for b4 being non empty transitive directed NetStr over b1 holds
           b3 . lim_inf b4 <= lim_inf (b3 * b4)
   holds b3 is monotone(b1, b2);

:: WAYBEL17:sch 3
scheme WAYBEL17:sch 3
{F1 -> non empty TopRelStr,
  F2 -> non empty TopRelStr,
  F3 -> non empty TopRelStr,
  F4 -> Element of the carrier of F3(),
  F5 -> Relation-like Function-like set}:
F5() .: {F4(b1) where b1 is Element of the carrier of F1(): P1[b1]} = {F5() . F4(b1) where b1 is Element of the carrier of F2(): P1[b1]}
provided
   the carrier of F3() c= proj1 F5()
and
   the carrier of F1() = the carrier of F2();


:: WAYBEL17:th 12
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded continuous RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st b3 is directed-sups-preserving(b1, b2)
for b4 being Element of the carrier of b1 holds
   b3 . b4 = "\/"({b3 . b5 where b5 is Element of the carrier of b1: b5 is_way_below b4},b2);

:: WAYBEL17:th 13
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded up-complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st for b4 being Element of the carrier of b1 holds
           b3 . b4 = "\/"({b3 . b5 where b5 is Element of the carrier of b1: b5 is_way_below b4},b2)
   holds b3 is monotone(b1, b2);

:: WAYBEL17:th 14
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded up-complete RelStr
for b2 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded continuous RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st for b4 being Element of the carrier of b1 holds
        b3 . b4 = "\/"({b3 . b5 where b5 is Element of the carrier of b1: b5 is_way_below b4},b2)
for b4 being Element of the carrier of b1
for b5 being Element of the carrier of b2 holds
      b5 is_way_below b3 . b4
   iff
      ex b6 being Element of the carrier of b1 st
         b6 is_way_below b4 & b5 is_way_below b3 . b6;

:: WAYBEL17:th 15
theorem
for b1, b2 being non empty RelStr
for b3 being Element of bool the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st (ex_sup_of b3,b1 & ex_sup_of b4 .: b3,b2 or b1 is complete & b1 is antisymmetric & b2 is complete & b2 is antisymmetric) &
         b4 is monotone(b1, b2)
   holds "\/"(b4 .: b3,b2) <= b4 . "\/"(b3,b1);

:: WAYBEL17:th 16
theorem
for b1, b2 being non empty reflexive antisymmetric RelStr
for b3 being non empty directed Element of bool the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st (ex_sup_of b3,b1 & ex_sup_of b4 .: b3,b2 or b1 is up-complete & b2 is up-complete) &
         b4 is monotone(b1, b2)
   holds "\/"(b4 .: b3,b2) <= b4 . "\/"(b3,b1);

:: WAYBEL17:th 17
theorem
for b1, b2 being non empty RelStr
for b3 being Element of bool the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st (ex_inf_of b3,b1 & ex_inf_of b4 .: b3,b2 or b1 is complete & b1 is antisymmetric & b2 is complete & b2 is antisymmetric) &
         b4 is monotone(b1, b2)
   holds b4 . "/\"(b3,b1) <= "/\"(b4 .: b3,b2);

:: WAYBEL17:th 18
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima up-complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being non empty monotone NetStr over b1
      st b3 is monotone(b1, b2)
   holds b3 * b4 is monotone(b2);

:: WAYBEL17:funcreg 8
registration
  let a1, a2 be reflexive transitive antisymmetric with_suprema with_infima up-complete RelStr;
  let a3 be Function-like quasi_total monotone Relation of the carrier of a1,the carrier of a2;
  let a4 be non empty monotone NetStr over a1;
  cluster a3 * a4 -> strict monotone;
end;

:: WAYBEL17:th 19
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima up-complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st for b4 being non empty transitive directed NetStr over b1 holds
        b3 . lim_inf b4 <= lim_inf (b3 * b4)
for b4 being non empty directed Element of bool the carrier of b1 holds
   "\/"(b3 .: b4,b2) = b3 . "\/"(b4,b1);

:: WAYBEL17:th 20
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being non empty transitive directed NetStr over b1
for b5 being Element of the carrier of b4
for b6 being Element of the carrier of b3 * b4
      st b6 = b5 & b3 is monotone(b1, b2)
   holds b3 . "/\"({b4 . b7 where b7 is Element of the carrier of b4: b5 <= b7},b1) <= "/\"({(b3 * b4) . b7 where b7 is Element of the carrier of b3 * b4: b6 <= b7},b2);

:: WAYBEL17:th 21
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is continuous(b1, b2)
   iff
      for b4 being non empty transitive directed NetStr over b1 holds
         b3 . lim_inf b4 <= lim_inf (b3 * b4);

:: WAYBEL17:th 22
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
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 directed-sups-preserving(b1, b2);

:: WAYBEL17:condreg 3
registration
  let a1, a2 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr;
  cluster Function-like quasi_total continuous -> directed-sups-preserving (Relation of the carrier of a1,the carrier of a2);
end;

:: WAYBEL17:condreg 4
registration
  let a1, a2 be TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr;
  cluster Function-like quasi_total directed-sups-preserving -> continuous (Relation of the carrier of a1,the carrier of a2);
end;

:: WAYBEL17:th 23
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Scott TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is continuous(b1, b2)
   iff
      for b4 being Element of the carrier of b1
      for b5 being Element of the carrier of b2 holds
            b5 is_way_below b3 . b4
         iff
            ex b6 being Element of the carrier of b1 st
               b6 is_way_below b4 & b5 is_way_below b3 . b6;

:: WAYBEL17:th 24
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Scott TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is continuous(b1, b2)
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = "\/"({b3 . b5 where b5 is Element of the carrier of b1: b5 is_way_below b4},b2);

:: WAYBEL17:th 25
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st for b4 being Element of the carrier of b1 holds
           b3 . b4 = "\/"({b3 . b5 where b5 is Element of the carrier of b1: b5 <= b4 & b5 is compact(b1)},b2)
   holds b3 is monotone(b1, b2);

:: WAYBEL17:th 26
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
   st for b4 being Element of the carrier of b1 holds
        b3 . b4 = "\/"({b3 . b5 where b5 is Element of the carrier of b1: b5 <= b4 & b5 is compact(b1)},b2)
for b4 being Element of the carrier of b1 holds
   b3 . b4 = "\/"({b3 . b5 where b5 is Element of the carrier of b1: b5 is_way_below b4},b2);

:: WAYBEL17:th 27
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b1 is algebraic & b2 is algebraic
   holds    b3 is continuous(b1, b2)
   iff
      for b4 being Element of the carrier of b1
      for b5 being Element of the carrier of b2
            st b5 in the carrier of CompactSublatt b2
         holds    b5 <= b3 . b4
         iff
            ex b6 being Element of the carrier of b1 st
               b6 in the carrier of CompactSublatt b1 & b6 <= b4 & b5 <= b3 . b6;

:: WAYBEL17:th 28
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b1 is algebraic & b2 is algebraic
   holds    b3 is continuous(b1, b2)
   iff
      for b4 being Element of the carrier of b1 holds
         b3 . b4 = "\/"({b3 . b5 where b5 is Element of the carrier of b1: b5 <= b4 & b5 is compact(b1)},b2);

:: WAYBEL17:th 29
theorem
for b1, b2 being non empty TopSpace-like reflexive transitive antisymmetric up-complete Scott TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is directed-sups-preserving(b1, b2)
   holds b3 is continuous(b1, b2);