formal/mizar/altcat_1.miz

:: Categories without Uniqueness of { \bf cod } and { \bf dom }
::  by Andrzej Trybulec

environ

 vocabularies FUNCT_1, SUBSET_1, RELAT_1, FUNCT_2, XBOOLE_0, TARSKI, PBOOLE,
      ZFMISC_1, MCART_1, FUNCOP_1, STRUCT_0, CAT_1, RELAT_2, BINOP_1, CARD_1,
      ALTCAT_1;
 notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XTUPLE_0, MCART_1, DOMAIN_1,
      RELAT_1, CARD_1, ORDINAL1, NUMBERS, RELSET_1, STRUCT_0, FUNCT_1, FUNCT_2,
      BINOP_1, MULTOP_1, FUNCOP_1, PBOOLE;
 constructors PARTFUN1, BINOP_1, MULTOP_1, PBOOLE, REALSET2, RELSET_1,
      XTUPLE_0;
 registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2, FUNCOP_1,
      STRUCT_0, RELSET_1, ZFMISC_1, CARD_1, XTUPLE_0;
 requirements BOOLE, SUBSET;
 definitions TARSKI, STRUCT_0, FUNCOP_1;
 equalities TARSKI, FUNCOP_1, BINOP_1, XTUPLE_0, ORDINAL1;
 expansions TARSKI;
 theorems FUNCT_1, ZFMISC_1, PBOOLE, DOMAIN_1, MULTOP_1, MCART_1, FUNCT_2,
      TARSKI, FUNCOP_1, RELAT_1, STRUCT_0, RELSET_1, XBOOLE_1, PARTFUN1,
      XTUPLE_0, FUNCT_4;
 schemes FUNCT_1;

begin :: Preliminaries

reserve i,j,k,x for object;

::$CT 4

theorem
  for A,B being set, F being ManySortedSet of [:B,A:], C being Subset of
A, D being Subset of B, x,y being set st x in C & y in D holds F.(y,x) = (F|([:
  D,C:] qua set)).(y,x) by FUNCT_1:49,ZFMISC_1:87;

scheme
  MSSLambda2{ A,B() -> set, F(object,object) -> object }:
 ex M being ManySortedSet of [:A(),B():]
 st for i,j being set st i in A() & j in B() holds M.(i,j) = F(i,j) proof
  deffunc F(object) = F($1`1,$1`2);
  consider f being Function such that
A1: dom f = [:A(),B():] and
A2: for x being object st x in [:A(),B():] holds f.x = F(x)
   from FUNCT_1:sch 3;
  reconsider f as ManySortedSet of [:A(),B():] by A1,PARTFUN1:def 2
,RELAT_1:def 18;
  take f;
  let i,j be set;
  assume i in A() & j in B();
  then
A3: [i,j] in [:A(),B():] by ZFMISC_1:87;
  [i,j]`1 = i & [i,j]`2 = j;
  hence thesis by A2,A3;
end;

scheme
  MSSLambda2D{ A,B() -> non empty set, F(object,object) -> object }:
 ex M being
ManySortedSet of [:A(),B():] st for i being Element of A(), j being Element of
  B() holds M.(i,j) = F(i,j) proof
  consider M being ManySortedSet of [:A(),B():] such that
A1: for i,j being set st i in A() & j in B() holds M.(i,j) = F(i,j) from
  MSSLambda2;
  take M;
  thus thesis by A1;
end;

scheme
  MSSLambda3{ A,B,C() -> set, F(object,object,object) -> object }:
 ex M being ManySortedSet of [:A(),B(),C():] st
 for i,j,k being set st i in A() & j in B() & k in C()
  holds M.(i,j,k) = F(i,j,k) proof
  deffunc F(object) = F($1`1`1,$1`1`2,$1`2);
  consider f being Function such that
A1: dom f = [:A(),B(),C():] and
A2: for x being object st x in [:A(),B(),C():] holds f.x = F(x)
   from FUNCT_1:sch 3;
  reconsider f as ManySortedSet of [:A(),B(),C():] by A1,PARTFUN1:def 2
,RELAT_1:def 18;
  take f;
  let i,j,k be set;
A3: [[i,j],k]`2 = k & [i,j,k] = [[i,j],k];
A4: [[i,j],k]`1`2 = j;
A5: [[i,j],k]`1`1 = i;
  assume i in A() & j in B() & k in C();
  then
A6: [i,j,k] in [:A(),B(),C():] by MCART_1:69;
  thus f.(i,j,k) = f.[i,j,k] by MULTOP_1:def 1
    .= F(i,j,k) by A2,A6,A5,A4,A3;
end;

scheme
  MSSLambda3D{ A,B,C() -> non empty set, F(object,object,object) -> object }:
ex M being
ManySortedSet of [:A(),B(),C():] st for i being Element of A(), j being Element
  of B(), k being Element of C() holds M.(i,j,k) = F(i,j,k) proof
  consider M being ManySortedSet of [:A(),B(),C():] such that
A1: for i,j,k being set
 st i in A() & j in B() & k in C() holds M.(i,j,k) = F(i,j,
  k) from MSSLambda3;
  take M;
  thus thesis by A1;
end;

theorem Th2:
  for A,B being set, N,M being ManySortedSet of [:A,B:] st for i,j
  st i in A & j in B holds N.(i,j) = M.(i,j) holds M = N
proof
  let A,B be set, N,M be ManySortedSet of [:A,B:];
  assume
A1: for i,j st i in A & j in B holds N.(i,j) = M.(i,j);
A2: now
    let x be object;
    assume
A3: x in [:A,B:];
    then reconsider A1 = A, B1 = B as non empty set;
    consider i being Element of A1, j being Element of B1 such that
A4: x = [i,j] by A3,DOMAIN_1:1;
    thus N.x = N.(i,j) by A4
      .= M.(i,j) by A1
      .= M.x by A4;
  end;
  dom M = [:A,B:] & dom N = [:A,B:] by PARTFUN1:def 2;
  hence thesis by A2,FUNCT_1:2;
end;

theorem Th3:
  for A,B being non empty set, N,M being ManySortedSet of [:A,B:]
st for i being Element of A, j being Element of B holds N.(i,j) = M.(i,j) holds
  M = N
proof
  let A,B be non empty set, N,M be ManySortedSet of [:A,B:];
  assume
  for i being Element of A, j being Element of B holds N.(i,j) = M.(i, j);
  then for i,j st i in A & j in B holds N.(i,j) = M.(i,j);
  hence thesis by Th2;
end;

theorem Th4:
  for A being set, N,M being ManySortedSet of [:A,A,A:] st for i,j
  ,k st i in A & j in A & k in A holds N.(i,j,k) = M.(i,j,k) holds M = N
proof
  let A be set, N,M be ManySortedSet of [:A,A,A:];
  assume
A1: for i,j,k st i in A & j in A & k in A holds N.(i,j,k) = M.(i,j,k);
A2: now
    let x be object;
    assume
A3: x in [:A,A,A:];
    then reconsider A as non empty set by MCART_1:31;
    consider i,j,k being Element of A such that
A4: x = [i,j,k] by A3,DOMAIN_1:3;
    thus M.x = M.(i,j,k) by A4,MULTOP_1:def 1
      .= N.(i,j,k) by A1
      .= N.x by A4,MULTOP_1:def 1;
  end;
  dom M = [:A,A,A:] & dom N = [:A,A,A:] by PARTFUN1:def 2;
  hence thesis by A2,FUNCT_1:2;
end;

begin :: Alternative Graphs

definition
  struct(1-sorted) AltGraph (# carrier -> set, Arrows -> ManySortedSet of [:
    the carrier, the carrier:] #);
end;

definition
  let G be AltGraph;
  mode Object of G is Element of G;
end;

definition
  let G be AltGraph;
  let o1,o2 be Object of G;

  func <^o1,o2^> -> set equals
  (the Arrows of G).(o1,o2);
  correctness;
end;

definition
  let G be AltGraph;
  let o1,o2 be Object of G;
  mode Morphism of o1,o2 is Element of <^o1,o2^>;
end;

definition
  let G be AltGraph;

  attr G is transitive means
  :Def2:
  for o1,o2,o3 being Object of G st <^o1,o2
  ^> <> {} & <^o2,o3^> <> {} holds <^o1,o3^> <> {};
end;

begin :: Binary Compositions

definition
  let I be set;
  let G be ManySortedSet of [:I,I:];
  func {|G|} -> ManySortedSet of [:I,I,I:] means
  :Def3:
  for i,j,k being object st i in I & j in I & k in I
   holds it.(i,j,k) = G.(i,k);
  existence
  proof
    deffunc F(object,object,object) = G.($1,$3);
    consider M being ManySortedSet of [:I,I,I:] such that
A1:   for i,j,k being set st i in I & j in I
    & k in I holds M.(i,j,k) = F(i,j,k) from MSSLambda3;
    take M;
    let i,j,k be object;
    thus thesis by A1;
  end;
  uniqueness
  proof
    let M1,M2 be ManySortedSet of [:I,I,I:] such that
A2: for i,j,k being object
st i in I & j in I & k in I holds M1.(i,j,k) = G.(i,k) and
A3: for i,j,k being object
st i in I & j in I & k in I holds M2.(i,j,k) = G.(i,k);
    now
      let i,j,k;
      assume
A4:   i in I & j in I & k in I;
      hence M1.(i,j,k) = G.(i,k) by A2
        .= M2.(i,j,k) by A3,A4;
    end;
    hence M1 = M2 by Th4;
  end;
  let H be ManySortedSet of [:I,I:];
  func {|G,H|} -> ManySortedSet of [:I,I,I:] means
  :Def4:
  for i,j,k being object st i in I & j in I & k in I
   holds it.(i,j,k) = [:H.(j,k),G.(i,j):];
  existence
  proof
    deffunc F(object,object,object) = [:H.($2,$3),G.($1,$2):];
    consider M being ManySortedSet of [:I,I,I:] such that
A5:   for i,j,k being set st i in I & j in I & k in I
    holds M.(i,j,k) = F(i,j,k)
       from MSSLambda3;
    take M;
    let i,j,k be object;
    thus thesis by A5;
  end;
  uniqueness
  proof
    let M1,M2 be ManySortedSet of [:I,I,I:] such that
A6: for i,j,k being object
     st i in I & j in I & k in I holds M1.(i,j,k) = [:H.(j,k)
    ,G.(i,j):] and
A7: for i,j,k being object
    st i in I & j in I & k in I holds M2.(i,j,k) = [:H.(j,k)
    ,G.(i,j):];
    now
      let i,j,k;
      assume
A8:   i in I & j in I & k in I;
      hence M1.(i,j,k) = [:H.(j,k),G.(i,j):] by A6
        .= M2.(i,j,k) by A7,A8;
    end;
    hence M1 = M2 by Th4;
  end;
end;

definition
  let I be set;
  let G be ManySortedSet of [:I,I:];
  mode BinComp of G is ManySortedFunction of {|G,G|},{|G|};
end;

definition
  let I be non empty set;
  let G be ManySortedSet of [:I,I:];
  let o be BinComp of G;
  let i,j,k be Element of I;
  redefine func o.(i,j,k) -> Function of [:G.(j,k),G.(i,j):], G.(i,k);
  coherence
  proof
A1: {|G|}.[i,j,k] = {|G|}.(i,j,k) by MULTOP_1:def 1
      .= G.(i,k) by Def3;
A2: o.[i,j,k] = o.(i,j,k) by MULTOP_1:def 1;
    {|G,G|}.[i,j,k] = {|G,G|}.(i,j,k) by MULTOP_1:def 1
      .= [:G.(j,k),G.(i,j):] by Def4;
    hence thesis by A2,A1,PBOOLE:def 15;
  end;
end;

definition
  let I be non empty set;
  let G be ManySortedSet of [:I,I:];
  let IT be BinComp of G;
  attr IT is associative means

  for i,j,k,l being Element of I for f,g,h
being set st f in G.(i,j) & g in G.(j,k) & h in G.(k,l) holds IT.(i,k,l).(h,IT.
  (i,j,k).(g,f)) = IT.(i,j,l).(IT.(j,k,l).(h,g),f);
  attr IT is with_right_units means

  for i being Element of I ex e being
set st e in G.(i,i) & for j being Element of I, f be set st f in G.(i,j) holds
  IT.(i,i,j).(f,e) = f;
  attr IT is with_left_units means

  for j being Element of I ex e being
set st e in G.(j,j) & for i being Element of I, f be set st f in G.(i,j) holds
  IT.(i,j,j).(e,f) = f;
end;

begin :: Alternative categories

definition
  struct(AltGraph) AltCatStr (# carrier -> set, Arrows -> ManySortedSet of [:
    the carrier, the carrier:], Comp -> BinComp of the Arrows #);
end;

registration
  cluster strict non empty for AltCatStr;
  existence
  proof

set X = the non empty set,A = the ManySortedSet of [:X,X:],C = the BinComp of A
;
    take AltCatStr(#X,A,C#);
    thus AltCatStr(#X,A,C#) is strict;
    thus the carrier of AltCatStr(#X,A,C#) is non empty;
  end;
end;

definition
  let C be non empty AltCatStr;
  let o1,o2,o3 be Object of C such that
A1: <^o1,o2^> <> {} & <^o2,o3^> <> {};
  let f be Morphism of o1,o2, g be Morphism of o2,o3;
  func g*f -> Morphism of o1,o3 equals
  :Def8:
  (the Comp of C).(o1,o2,o3).(g,f
  );
  coherence
  proof
    reconsider H1 = <^o1,o2^>, H2 = <^o2,o3^> as non empty set by A1;
    reconsider F = (the Comp of C).(o1,o2,o3) as Function of [:H2, H1:], <^o1,
    o3^>;
    reconsider y = g as Element of H2;
    reconsider x = f as Element of H1;
    F.(y,x) is Element of <^o1,o3^>;
    hence thesis;
  end;
  correctness;
end;

definition
  let IT be Function;
  attr IT is compositional means
  :Def9:
  x in dom IT implies ex f,g being Function st x = [g,f] & IT.x = g*f;
end;

registration
  let A,B be functional set;
  cluster compositional for ManySortedFunction of [:A,B:];
  existence
  proof
    per cases;
    suppose
A1:   A = {} or B = {};
      set M = EmptyMS[:A,B:];
      M is Function-yielding by A1;
      then reconsider M as ManySortedFunction of [:A,B:];
      take M;
      let x;
      thus thesis by A1;
    end;
    suppose
      A <> {} & B <> {};
      then reconsider A1=A, B1=B as non empty functional set;
      deffunc F(Element of A1,Element of B1) = $1*$2;
      consider M being ManySortedSet of [:A1,B1:] such that
A2:   for i being Element of A1, j being Element of B1 holds M.(i,j) =
      F(i,j) from MSSLambda2D;
      M is Function-yielding
      proof
        let x be object;
        assume x in dom M;
        then
A3:     x in [:A1,B1:];
        then
A4:     x`1 in A1 & x `2 in B1 by MCART_1:10;
        then reconsider f = x`1, g = x`2 as Function;
        M.x = M.(f,g) by A3,MCART_1:22
          .= f*g by A2,A4;
        hence thesis;
      end;
      then reconsider M as ManySortedFunction of [:A,B:];
      take M;
      let x;
      assume x in dom M;
      then
A5:   x in [:A1,B1:];
      then
A6:   x`1 in A1 & x `2 in B1 by MCART_1:10;
      then reconsider f = x`1, g = x`2 as Function;
      take g,f;
      thus x = [f,g] by A5,MCART_1:22;
      thus M.x = M.(f,g) by A5,MCART_1:22
        .= f*g by A2,A6;
    end;
  end;
end;

::$CT 2

theorem Th5:
  for A,B being functional set for F being compositional
ManySortedSet of [:A,B:], g,f being Function st g in A & f in B holds F.(g,f) =
  g*f
proof
  let A,B be functional set;
  let F be compositional ManySortedSet of [:A,B:], g,f be Function such that
A1: g in A & f in B;
  dom F = [:A,B:] by PARTFUN1:def 2;
  then [g,f] in dom F by A1,ZFMISC_1:87;
  then consider ff,gg being Function such that
A2: [g,f] = [gg,ff] and
A3: F.[g,f] = gg*ff by Def9;
  g = gg by A2,XTUPLE_0:1;
  hence thesis by A2,A3,XTUPLE_0:1;
end;

definition
  let A,B be functional set;
  func FuncComp(A,B) -> compositional ManySortedFunction of [:B,A:] means
:Def10: not contradiction;
  uniqueness
  proof
    let M1,M2 be compositional ManySortedFunction of [:B,A:];
    now
      let i,j;
      assume i in B & j in A;
      then
A1:   [i,j] in [:B,A:] by ZFMISC_1:87;
      then [i,j] in dom M1 by PARTFUN1:def 2;
      then consider f1,g1 being Function such that
A2:   [i,j] = [g1,f1] and
A3:   M1.[i,j] = g1*f1 by Def9;
      [i,j] in dom M2 by A1,PARTFUN1:def 2;
      then consider f2,g2 being Function such that
A4:   [i,j] = [g2,f2] and
A5:   M2.[i,j] = g2*f2 by Def9;
      g1 = g2 by A2,A4,XTUPLE_0:1;
      hence M1.(i,j) = M2.(i,j) by A2,A3,A4,A5,XTUPLE_0:1;
    end;
    hence M1 = M2 by Th2;
  end;
  correctness;
end;

theorem Th6:
  for A,B,C being set holds rng FuncComp(Funcs(A,B),Funcs(B,C)) c= Funcs(A,C)
proof
  let A,B,C be set;
  let i be object;
  set F = FuncComp(Funcs(A,B),Funcs(B,C));
  assume i in rng F;
  then consider j being object such that
A1: j in dom F and
A2: i = F.j by FUNCT_1:def 3;
  consider f,g being Function such that
A3: j = [g,f] and
A4: F.j = g*f by A1,Def9;
  g in Funcs(B,C) & f in Funcs(A,B) by A1,A3,ZFMISC_1:87;
  hence thesis by A2,A4,FUNCT_2:128;
end;

theorem Th7:
  for o be set holds FuncComp({id o},{id o}) = (id o,id o) :-> id o
proof
  let o be set;
A1: dom FuncComp({id o},{id o}) = [:{id o},{id o}:] by PARTFUN1:def 2;
  rng FuncComp({id o},{id o}) c= {id o}
  proof
    let i be object;
    assume i in rng FuncComp({id o},{id o});
    then consider j being object such that
A2: j in [:{id o},{id o}:] and
A3: i = FuncComp({id o},{id o}).j by A1,FUNCT_1:def 3;
    consider f,g being Function such that
A4: j = [g,f] and
A5: FuncComp({id o},{id o}).j = g*f by A1,A2,Def9;
    f in {id o} by A2,A4,ZFMISC_1:87;
    then
A6: f = id o by TARSKI:def 1;
    g in {id o} by A2,A4,ZFMISC_1:87;
    then o /\ o = o & g = id o by TARSKI:def 1;
    then i = id o by A3,A5,A6,FUNCT_1:22;
    hence thesis by TARSKI:def 1;
  end;
  then FuncComp({id o},{id o}) is Function of [:{id o},{id o}:],{id o} by A1,
RELSET_1:4;
  hence thesis by FUNCOP_1:def 10;
end;

theorem Th8:
  for A,B being functional set, A1 being Subset of A, B1 being
  Subset of B holds FuncComp(A1,B1) = FuncComp(A,B)|([:B1,A1:] qua set)
proof
  let A,B be functional set, A1 be Subset of A, B1 be Subset of B;
  set f = FuncComp(A,B)|([:B1,A1:] qua set);
A1: dom FuncComp(A,B) = [:B,A:] by PARTFUN1:def 2;
  then
A2: dom f = [:B1,A1:] by RELAT_1:62;
  then reconsider f as ManySortedFunction of [:B1,A1:] by PARTFUN1:def 2;
  f is compositional
  proof
    let i;
    assume
A3: i in dom f;
    then f.i = FuncComp(A,B).i by FUNCT_1:49;
    hence thesis by A1,A2,A3,Def9;
  end;
  hence thesis by Def10;
end;

:: Kategorie przeksztalcen, Semadeni Wiweger, 1.2.2, str.15

definition
  let C be non empty AltCatStr;
  attr C is quasi-functional means

  for a1,a2 being Object of C holds <^a1,a2^> c= Funcs(a1,a2);
  attr C is semi-functional means

  for a1,a2,a3 being Object of C st <^
a1,a2^> <> {} & <^a2,a3^> <> {} & <^a1,a3^> <> {} for f being Morphism of a1,a2
, g being Morphism of a2,a3, f9,g9 being Function st f = f9 & g = g9 holds g*f
  =g9*f9;
  attr C is pseudo-functional means
  :Def13:
  for o1,o2,o3 being Object of C
holds (the Comp of C).(o1,o2,o3) = FuncComp(Funcs(o1,o2),Funcs(o2,o3))|([:<^o2,
  o3^>,<^o1,o2^>:] qua set);
end;

registration
  let X be non empty set, A be ManySortedSet of [:X,X:], C be BinComp of A;
  cluster AltCatStr(#X,A,C#) -> non empty;
  coherence;
end;

registration
  cluster strict pseudo-functional for non empty AltCatStr;
  existence
  proof
A1: {[0,0,0]} = [: {0},{0},{0} :] by MCART_1:35;
    then reconsider c = [0,0,0] .--> FuncComp(Funcs(0,0),Funcs(0,0)) as
    ManySortedSet of [: {0},{0},{0} :];
    reconsider c as ManySortedFunction of [: {0},{0},{0} :];
    dom([0,0] .--> Funcs(0,0)) = {[0,0]}
      .= [: {0},{0} :] by ZFMISC_1:29;
    then reconsider
    m = [0,0] .--> Funcs(0,0) as ManySortedSet of [: {0},{0} :];
A2: m.(0,0) = Funcs(0,0) by FUNCOP_1:72;
A3: 0 in {0} by TARSKI:def 1;
    now
      let i;
      reconsider ci = c.i as Function;
      assume i in [: {0},{0},{0} :];
      then
A4:   i = [0,0,0] by A1,TARSKI:def 1;
      then
A5:   c.i = FuncComp(Funcs(0,0),Funcs(0,0)) by FUNCOP_1:72;
      then
A6:   dom ci = [:m.(0,0),m.(0,0):] by A2,PARTFUN1:def 2
        .= {|m,m|}.(0,0,0) by A3,Def4
        .= {|m,m|}.i by A4,MULTOP_1:def 1;
A7:   {|m|}.i = {|m|}.(0,0,0) by A4,MULTOP_1:def 1
        .= m.(0,0) by A3,Def3;
      then rng ci c= {|m|}.i by A2,A5,Th6;
      hence c.i is Function of {|m,m|}.i, {|m|}.i by A2,A6,A7,FUNCT_2:def 1
,RELSET_1:4;
    end;
    then reconsider c as BinComp of m by PBOOLE:def 15;
    take C = AltCatStr(#{0},m,c#);
    thus C is strict;
    let o1,o2,o3 be Object of C;
A8: o3 = 0 by TARSKI:def 1;
A9: o1 = 0 & o2 = 0 by TARSKI:def 1;
    then
A10: dom FuncComp(Funcs(0,0),Funcs(0,0)) = [:<^o2,o3^>,<^o1,o2^>:] by A2,A8,
PARTFUN1:def 2;
    thus (the Comp of C).(o1,o2,o3) = c.[o1,o2,o3] by MULTOP_1:def 1
      .= FuncComp(Funcs(0,0),Funcs(0,0)) by A9,A8,FUNCOP_1:72
      .= FuncComp(Funcs(o1,o2),Funcs(o2,o3))|([:<^o2,o3^>,<^o1,o2^>:] qua
    set) by A9,A8,A10;
  end;
end;

theorem
  for C being non empty AltCatStr, a1,a2,a3 being Object of C holds (the
  Comp of C).(a1,a2,a3) is Function of [:<^a2,a3^>,<^a1,a2^>:],<^a1,a3^>;

theorem Th10:
  for C being pseudo-functional non empty AltCatStr for a1,a2,a3
being Object of C st <^a1,a2^> <> {} & <^a2,a3^> <> {} & <^a1,a3^> <> {} for f
being Morphism of a1,a2, g being Morphism of a2,a3, f9,g9 being Function st f =
  f9 & g = g9 holds g*f =g9*f9
proof
  let C be pseudo-functional non empty AltCatStr;
  let a1,a2,a3 be Object of C such that
A1: <^a1,a2^> <> {} & <^a2,a3^> <> {} and
A2: <^a1,a3^> <> {};
  let f be Morphism of a1,a2, g be Morphism of a2,a3, f9,g9 be Function such
  that
A3: f = f9 & g = g9;
A4: [g,f] in [:<^a2,a3^>,<^a1,a2^>:] by A1,ZFMISC_1:87;
A5: (the Comp of C).(a1,a2,a3) = FuncComp(Funcs(a1,a2),Funcs(a2,a3))|([:<^a2
  ,a3^>,<^a1,a2^>:] qua set) by Def13;
  dom(FuncComp(Funcs(a1,a2),Funcs(a2,a3))|([:<^a2,a3^>,<^a1,a2^>:] qua set
)) c= dom(FuncComp(Funcs(a1,a2),Funcs(a2,a3))) & dom((the Comp of C).(a1,a2,a3)
  ) = [:<^a2,a3^>,<^a1,a2^>:] by A2,FUNCT_2:def 1,RELAT_1:60;
  then consider f99,g99 being Function such that
A6: [g,f] = [g99,f99] and
A7: FuncComp(Funcs(a1,a2),Funcs(a2,a3)).[g,f] = g99*f99 by A5,A4,Def9;
A8: g = g99 & f = f99 by A6,XTUPLE_0:1;
  thus g*f = (the Comp of C).(a1,a2,a3).(g,f) by A1,Def8
    .= g9*f9 by A5,A3,A4,A7,A8,FUNCT_1:49;
end;

:: Kategorie EnsCat(A), Semadeni Wiweger 1.2.3, str. 15
:: ale bez parametryzacji

definition
  let A be non empty set;
  func EnsCat A -> strict pseudo-functional non empty AltCatStr means
  :Def14
  :
  the carrier of it = A & for a1,a2 being Object of it holds <^a1,a2^> = Funcs(
  a1,a2);
  existence
  proof
    deffunc F(set,set,set) = FuncComp(Funcs($1,$2),Funcs($2,$3));
    consider M being ManySortedSet of [:A,A:] such that
A1: for i,j being set st i in A & j in A holds M.(i,j) = Funcs(i,j) from
    MSSLambda2;
    consider c being ManySortedSet of [:A,A,A:] such that
A2: for i,j,k being set st i in A & j in A & k in A
     holds c.(i,j,k) = F(i,j,k)
    from MSSLambda3;
    c is Function-yielding
    proof
      let i be object;
      assume i in dom c;
      then i in [:A,A,A:];
      then consider x1,x2,x3 being object such that
A3:   x1 in A & x2 in A & x3 in A and
A4:   i = [x1,x2,x3] by MCART_1:68;
      reconsider x1,x2,x3 as set by TARSKI:1;
      c.i = c.(x1,x2,x3) by A4,MULTOP_1:def 1
        .= FuncComp(Funcs(x1,x2),Funcs(x2,x3)) by A2,A3;
      hence thesis;
    end;
    then reconsider c as ManySortedFunction of [:A,A,A:];
    now
      let i;
      reconsider ci = c.i as Function;
      assume i in [:A,A,A:];
      then consider x1,x2,x3 being object such that
A5:   x1 in A and
A6:   x2 in A and
A7:   x3 in A and
A8:   i = [x1,x2,x3] by MCART_1:68;
A9:   {|M|}.i = {|M|}.(x1,x2,x3) by A8,MULTOP_1:def 1
        .= M.(x1,x3) by A5,A6,A7,Def3;
      reconsider x1,x2,x3 as set by TARSKI:1;
A10:  c.i = c.(x1,x2,x3) by A8,MULTOP_1:def 1
        .= FuncComp(Funcs(x1,x2),Funcs(x2,x3)) by A2,A5,A6,A7;
      M.(x1,x2) = Funcs(x1,x2) & M.(x2,x3) = Funcs(x2,x3) by A1,A5,A6,A7;
      then
A11:  [:Funcs(x2,x3),Funcs(x1,x2):] = {|M,M|}.(x1,x2,x3) by A5,A6,A7,Def4
        .= {|M,M|}.i by A8,MULTOP_1:def 1;
      M.(x1,x3) = Funcs(x1,x3) by A1,A5,A7;
      then
A12:  rng ci c= {|M|}.i by A10,A9,Th6;
      dom ci = [:Funcs(x2,x3),Funcs(x1,x2):] by A10,PARTFUN1:def 2;
      hence c.i is Function of {|M,M|}.i, {|M|}.i
      by A11,A12,FUNCT_2:2;
    end;
    then reconsider c as BinComp of M by PBOOLE:def 15;
    set C = AltCatStr(#A,M,c#);
    C is pseudo-functional
    proof
      let o1,o2,o3 be Object of C;
      <^o1,o2^> = Funcs(o1,o2) & <^o2,o3^> = Funcs(o2,o3) by A1;
      then
A13:  dom FuncComp(Funcs(o1,o2),Funcs(o2,o3)) = [:<^o2,o3^>,<^o1,o2 ^> :]
      by PARTFUN1:def 2;
      thus (the Comp of C).(o1,o2,o3) = FuncComp(Funcs(o1,o2),Funcs(o2,o3)) by
A2
        .= FuncComp(Funcs(o1,o2),Funcs(o2,o3))|([:<^o2,o3^>,<^o1,o2^>:] qua
      set) by A13;
    end;
    then reconsider C as strict pseudo-functional non empty AltCatStr;
    take C;
    thus the carrier of C = A;
    let a1,a2 be Object of C;
    thus thesis by A1;
  end;
  uniqueness
  proof
    let C1,C2 be strict pseudo-functional non empty AltCatStr such that
A14: the carrier of C1 = A and
A15: for a1,a2 being Object of C1 holds <^a1,a2^> = Funcs(a1,a2) and
A16: the carrier of C2 = A and
A17: for a1,a2 being Object of C2 holds <^a1,a2^> = Funcs(a1,a2);
A18: now
      let i,j;
      assume
A19:  i in A & j in A;
      then reconsider a1 = i, a2 = j as Object of C1 by A14;
      reconsider b1 = i, b2 = j as Object of C2 by A16,A19;
      thus (the Arrows of C1).(i,j) = <^a1,a2^> .= Funcs(a1,a2) by A15
        .= <^b1,b2^> by A17
        .= (the Arrows of C2).(i,j);
    end;
A20: now
      let i,j,k;
      assume
A21:  i in A & j in A & k in A;
      then reconsider a1 = i, a2 = j, a3 = k as Object of C1 by A14;
      reconsider b1 = i, b2 = j, b3 = k as Object of C2 by A16,A21;
      <^a2,a3^> = <^b2,b3^> & <^a1,a2^> = <^b1,b2^> by A14,A18;
      hence (the Comp of C1).(i,j,k) = FuncComp(Funcs(b1,b2),Funcs(b2,b3))|([:
      <^b2,b3^>,<^b1,b2^>:] qua set) by Def13
        .= (the Comp of C2).(i,j,k) by Def13;
    end;
    the Arrows of C1 = the Arrows of C2 by A14,A16,A18,Th2;
    hence thesis by A14,A16,A20,Th4;
  end;
end;

definition
  let C be non empty AltCatStr;
  attr C is associative means
  :Def15:
  the Comp of C is associative;
  attr C is with_units means
  :Def16:
  the Comp of C is with_left_units with_right_units;
end;

Lm1: for A being non empty set holds EnsCat A is transitive associative
with_units
proof
  let A be non empty set;
  set G = the Arrows of EnsCat A, C = the Comp of EnsCat A;
  thus
A1: EnsCat A is transitive
  proof
    let o1,o2,o3 be Object of EnsCat A;
    assume <^o1,o2^> <> {} & <^o2,o3^> <> {};
    then Funcs(o1,o2) <> {} & Funcs(o2,o3) <> {} by Def14;
    then Funcs(o1,o3) <> {} by FUNCT_2:129;
    hence thesis by Def14;
  end;
  thus EnsCat A is associative
  proof
    let i,j,k,l be Element of EnsCat A;
    reconsider i9=i, j9=j, k9=k, l9 = l as Object of EnsCat A;
    let f,g,h be set such that
A2: f in G.(i,j) and
A3: g in G.(j,k) and
A4: h in G.(k,l);
    reconsider h99 = h as Morphism of k9,l9 by A4;
    reconsider g99 = g as Morphism of j9,k9 by A3;
A5: <^k9,l9^> = Funcs(k,l) by Def14;
    <^i9,j9^> = Funcs(i,j) & <^j9,k9^> = Funcs(j,k) by Def14;
    then reconsider f9 = f, g9 = g, h9 = h as Function by A2,A3,A4,A5;
A6: G.(k,l) = <^k9,l9^>;
A7: <^j9,k9^> <> {} by A3;
    then
A8: <^j9,l9^> <> {} by A1,A4,A6;
    then
A9: h99 * g99 = h9 * g9 by A3,A4,Th10;
    reconsider f99 = f as Morphism of i9,j9 by A2;
    G.(i,j) = <^i9,j9^>;
    then
A10: <^i9,k9^> <> {} by A1,A2,A7;
    then
A11: g99 * f99 = g9 * f9 by A2,A3,Th10;
A12: <^i9,l9^> <> {} by A1,A4,A6,A10;
A13: C.(j,k,l).(h,g) = h99 * g99 by A3,A4,Def8;
    C.(i,j,k).(g,f) = g99 * f99 by A2,A3,Def8;
    hence C.(i,k,l).(h,C.(i,j,k).(g,f)) = h99*(g99*f99) by A4,A10,Def8
      .= h9*(g9*f9) by A4,A10,A12,A11,Th10
      .= h9*g9*f9 by RELAT_1:36
      .= h99*g99*f99 by A2,A8,A12,A9,Th10
      .= C.(i,j,l).(C.(j,k,l).(h,g),f) by A2,A8,A13,Def8;
  end;
  thus the Comp of EnsCat A is with_left_units
  proof
    let i be Element of EnsCat A;
    reconsider i9 = i as Object of EnsCat A;
    take id i;
A14: <^i9,i9^> = Funcs(i,i) by Def14;
    hence id i in G.(i,i) by FUNCT_2:126;
    reconsider I = id i as Morphism of i9,i9 by A14,FUNCT_2:126;
    let j be Element of EnsCat A, f be set;
    reconsider j9 = j as Object of EnsCat A;
    assume
A15: f in G.(j,i);
    then reconsider f99 = f as Morphism of j9,i9;
    <^j9,i9^> = Funcs(j,i) by Def14;
    then reconsider f9 = f as Function of j,i by A15,FUNCT_2:66;
    thus C.(j,i,i).(id i,f) = I*f99 by A14,A15,Def8
      .= (id i)*f9 by A14,A15,Th10
      .= f by FUNCT_2:17;
  end;
  let i be Element of EnsCat A;
  reconsider i9 = i as Object of EnsCat A;
  take id i;
A16: <^i9,i9^> = Funcs(i,i) by Def14;
  hence id i in G.(i,i) by FUNCT_2:126;
  reconsider I = id i as Morphism of i9,i9 by A16,FUNCT_2:126;
  let j be Element of EnsCat A, f be set;
  reconsider j9 = j as Object of EnsCat A;
  assume
A17: f in G.(i,j);
  then reconsider f99 = f as Morphism of i9,j9;
  <^i9,j9^> = Funcs(i,j) by Def14;
  then reconsider f9 = f as Function of i,j by A17,FUNCT_2:66;
  thus C.(i,i,j).(f,id i) = f99*I by A16,A17,Def8
    .= f9*id i by A16,A17,Th10
    .= f by FUNCT_2:17;
end;

registration
  cluster transitive associative with_units strict for non empty AltCatStr;
  existence
  proof
    take EnsCat {{}};
    thus thesis by Lm1;
  end;
end;

theorem
  for C being transitive non empty AltCatStr, a1,a2,a3 being Object of C
holds dom((the Comp of C).(a1,a2,a3)) = [:<^a2,a3^>,<^a1,a2^>:] & rng((the Comp
  of C).(a1,a2,a3)) c= <^a1,a3^>
proof
  let C be transitive non empty AltCatStr, a1,a2,a3 be Object of C;
  <^a1,a3^> = {} implies <^a1,a2^> = {} or <^a2,a3^> = {} by Def2;
  then <^a1,a3^> = {} implies [:<^a2,a3^>,<^a1,a2^>:] = {};
  hence thesis by FUNCT_2:def 1,RELAT_1:def 19;
end;

theorem Th12:
  for C being with_units non empty AltCatStr for o being Object
  of C holds <^o,o^> <> {}
proof
  let C be with_units non empty AltCatStr;
  let o be Object of C;
  the Comp of C is with_left_units by Def16;
  then
  ex e being set st e in (the Arrows of C).(o,o) & for o9 being Element of
C, f be set st f in (the Arrows of C).(o9,o) holds (the Comp of C).(o9,o,o).(e,
  f) = f;
  hence thesis;
end;

registration
  let A be non empty set;
  cluster EnsCat A -> transitive associative with_units;
  coherence by Lm1;
end;

registration
  cluster quasi-functional semi-functional transitive -> pseudo-functional for

    non empty AltCatStr;
  coherence
  proof
    let C be non empty AltCatStr;
    assume
A1: C is quasi-functional semi-functional transitive;
    let o1,o2,o3 be Object of C;
    set c = (the Comp of C).(o1,o2,o3), f = FuncComp(Funcs(o1,o2),Funcs(o2,o3)
    )|([:<^o2,o3^>,<^o1,o2^>:] qua set);
    per cases;
    suppose
A2:   <^o2,o3^> = {} or <^o1,o2^> = {};
      hence c = {} .= f by A2;
    end;
    suppose
A3:   <^o2,o3^> <> {} & <^o1,o2^> <> {};
      then
A4:   <^o1,o3^> <> {} by A1;
      then
A5:   dom c = [:<^o2,o3^>,<^o1,o2^>:] by FUNCT_2:def 1;
A6:   <^o2,o3^> c= Funcs(o2,o3) & <^o1,o2^> c= Funcs(o1,o2) by A1;
      dom FuncComp(Funcs(o1,o2),Funcs(o2,o3)) = [:Funcs(o2,o3),Funcs(o1,o2
      ) :] by PARTFUN1:def 2;
      then
      dom f = [:Funcs(o2,o3),Funcs(o1,o2):] /\ [:<^o2,o3^>,<^o1,o2^> :] by
RELAT_1:61;
      then
A7:   dom c = dom f by A6,A5,XBOOLE_1:28,ZFMISC_1:96;
      now
        let i be object;
        assume
A8:     i in dom c;
        then consider i1,i2 being object such that
A9:     i1 in <^o2,o3^> and
A10:    i2 in <^o1,o2^> and
A11:    i = [i1,i2] by ZFMISC_1:84;
        reconsider a2 = i2 as Morphism of o1,o2 by A10;
        reconsider a1 = i1 as Morphism of o2,o3 by A9;
        reconsider g = i1, h = i2 as Function by A6,A9,A10;
        thus c.i = (the Comp of C).(o1,o2,o3).(a1,a2) by A11
          .= a1*a2 by A3,Def8
          .= g*h by A1,A3,A4
          .= FuncComp(Funcs(o1,o2),Funcs(o2,o3)).(g,h) by A6,A9,A10,Th5
          .= f.i by A7,A8,A11,FUNCT_1:47;
      end;
      hence thesis by A7,FUNCT_1:2;
    end;
  end;
  cluster with_units pseudo-functional transitive -> quasi-functional
    semi-functional for non empty AltCatStr;
  coherence
  proof
    let C be non empty AltCatStr such that
A12: C is with_units pseudo-functional transitive;
    thus C is quasi-functional
    proof
      let a1,a2 be Object of C;
      per cases;
      suppose
        <^a1,a2^> = {};
        hence thesis;
      end;
      suppose
A13:    <^a1,a2^> <> {};
        set c = (the Comp of C).(a1,a1,a2), f = FuncComp(Funcs(a1,a1),Funcs(a1
        ,a2));
A14:    dom c = [:<^a1,a2^>,<^a1,a1^>:] by A13,FUNCT_2:def 1;
        dom f = [:Funcs(a1,a2),Funcs(a1,a1):] & c = f|([:<^a1,a2^>,<^a1,
        a1^>:] qua set) by A12,PARTFUN1:def 2;
        then
A15:    [:<^a1,a2^>,<^a1,a1^>:] c= [:Funcs(a1,a2),Funcs(a1,a1):] by A14,
RELAT_1:60;
        <^a1,a1^> <> {} by A12,Th12;
        hence thesis by A15,ZFMISC_1:114;
      end;
    end;
    let a1,a2,a3 be Object of C;
    thus thesis by A12,Th10;
  end;
end;

:: Definicja kategorii, Semadeni Wiweger 1.3.1, str. 16-17

definition
  mode category is transitive associative with_units non empty AltCatStr;
end;

begin

definition
  let C be with_units non empty AltCatStr;
  let o be Object of C;
  func idm o -> Morphism of o,o means
  :Def17:
  for o9 being Object of C st <^o,
  o9^> <> {} for a being Morphism of o,o9 holds a*it = a;
  existence
  proof
    the Comp of C is with_right_units by Def16;
    then consider e being set such that
A1: e in (the Arrows of C).(o,o) and
A2: for o9 being Element of C, f be set st f in (the Arrows of C).(o,
    o9) holds (the Comp of C).(o,o,o9).(f,e) = f;
    reconsider e as Morphism of o,o by A1;
    take e;
    let o9 be Object of C such that
A3: <^o,o9^> <> {};
    let a be Morphism of o,o9;
    thus a*e = (the Comp of C).(o,o,o9).(a,e) by A1,A3,Def8
      .= a by A2,A3;
  end;
  uniqueness
  proof
    the Comp of C is with_left_units by Def16;
    then consider d being set such that
A4: d in (the Arrows of C).(o,o) and
A5: for o9 being Element of C, f be set st f in (the Arrows of C).(o9,
    o) holds (the Comp of C).(o9,o,o).(d,f) = f;
    reconsider d as Morphism of o,o by A4;
    let a1,a2 be Morphism of o,o such that
A6: for o9 being Object of C st <^o,o9^> <> {} for a being Morphism of
    o,o9 holds a*a1 = a and
A7: for o9 being Object of C st <^o,o9^> <> {} for a being Morphism of
    o,o9 holds a*a2 = a;
A8: <^o,o^> <> {} by Th12;
    hence a1 = (the Comp of C).(o,o,o).(d,a1) by A5
      .= d*a1 by A8,Def8
      .= d by A6,Th12
      .= d*a2 by A7,Th12
      .= (the Comp of C).(o,o,o).(d,a2) by A8,Def8
      .= a2 by A5,A8;
  end;
end;

theorem Th13:
  for C being with_units non empty AltCatStr for o being Object
  of C holds idm o in <^o,o^>
proof
  let C be with_units non empty AltCatStr;
  let o be Object of C;
  <^o,o^> <> {} by Th12;
  hence thesis;
end;

theorem
  for C being with_units non empty AltCatStr for o1,o2 being Object of
  C st <^o1,o2^> <> {} for a being Morphism of o1,o2 holds (idm o2)*a = a
proof
  let C be with_units non empty AltCatStr;
  let o1,o2 be Object of C such that
A1: <^o1,o2^> <> {};
  let a be Morphism of o1,o2;
  the Comp of C is with_left_units by Def16;
  then consider d being set such that
A2: d in (the Arrows of C).(o2,o2) and
A3: for o9 being Element of C, f be set st f in (the Arrows of C).(o9,o2
  ) holds (the Comp of C).(o9,o2,o2).(d,f) = f;
  reconsider d as Morphism of o2,o2 by A2;
  idm o2 in <^o2,o2^> by Th13;
  then d = d*idm o2 by Def17
    .= (the Comp of C).(o2,o2,o2).(d,idm o2) by A2,Def8
    .= idm o2 by A3,Th13;
  hence (idm o2)*a = (the Comp of C).(o1,o2,o2).(d,a) by A1,A2,Def8
    .= a by A1,A3;
end;

theorem
  for C being associative transitive non empty AltCatStr for o1,o2,o3,
o4 being Object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o4^> <> {} for
a being Morphism of o1,o2, b being Morphism of o2,o3, c being Morphism of o3,o4
  holds c*(b*a) = (c*b)*a
proof
  let C be associative transitive non empty AltCatStr;
  let o1,o2,o3,o4 be Object of C such that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {} and
A3: <^o3,o4^> <> {};
  let a be Morphism of o1,o2, b be Morphism of o2,o3, c be Morphism of o3,o4;
A4: <^o2,o4^> <> {} & c*b = (the Comp of C).(o2,o3,o4).(c,b) by A2,A3,Def2,Def8
;
A5: the Comp of C is associative by Def15;
  <^o1,o3^> <> {} & b*a = (the Comp of C).(o1,o2,o3).(b,a) by A1,A2,Def2,Def8;
  hence
  c*(b*a) = (the Comp of C).(o1,o3,o4).(c,(the Comp of C).(o1,o2,o3).(b,a
  )) by A3,Def8
    .= (the Comp of C).(o1,o2,o4).((the Comp of C).(o2,o3,o4).(c,b),a) by A1,A2
,A3,A5
    .= (c*b)*a by A1,A4,Def8;
end;

begin

:: kategoria dyskretna, Semadeni Wiweger, 1.3.1, str.18

definition
  let C be AltCatStr;
  attr C is quasi-discrete means
  :Def18:
  for i,j being Object of C st <^i,j^> <> {} holds i = j;

:: to sa po prostu zbiory monoidow
  attr C is pseudo-discrete means

  for i being Object of C holds <^i,i ^> is trivial;
end;

theorem
  for C being with_units non empty AltCatStr holds C is
  pseudo-discrete iff for o being Object of C holds <^o,o^> = { idm o }
proof
  let C be with_units non empty AltCatStr;
  hereby
    assume
A1: C is pseudo-discrete;
    let o be Object of C;
    now
      let j be object;
      hereby
        idm o in <^o,o^> & <^o,o^> is trivial by A1,Th13;
        then consider i being object such that
A2:     <^o,o^> = { i } by ZFMISC_1:131;
        assume j in <^o,o^>;
        then j = i by A2,TARSKI:def 1;
        hence j = idm o by A2,TARSKI:def 1;
      end;
      thus j = idm o implies j in <^o,o^> by Th13;
    end;
    hence <^o,o^> = { idm o } by TARSKI:def 1;
  end;
  assume
A3: for o be Object of C holds <^o,o^> = { idm o };
  let o be Object of C;
  <^o,o^> = { idm o } by A3;
  hence thesis;
end;

registration
  cluster 1-element -> quasi-discrete for AltCatStr;
  coherence
  by STRUCT_0:def 10;
end;

theorem Th17:
  EnsCat {0} is pseudo-discrete 1-element
proof
A1: the carrier of EnsCat {0} = {0} by Def14;
  hereby
    let i be Object of EnsCat {0};
    i = 0 by A1,TARSKI:def 1;
    hence <^i,i^> is trivial by Def14,FUNCT_2:127;
  end;
  thus the carrier of EnsCat {0} is 1-element by A1;
end;

registration
  cluster pseudo-discrete trivial strict for category;
  existence by Th17;
end;

registration
  cluster quasi-discrete pseudo-discrete trivial strict for category;
  existence by Th17;
end;

definition
  mode discrete_category is quasi-discrete pseudo-discrete category;
end;

definition
  let A be non empty set;
  func DiscrCat A -> quasi-discrete strict non empty AltCatStr means
  :Def20:
  the carrier of it = A & for i being Object of it holds <^i,i^> = { id i };
  existence
  proof
    deffunc F(Element of A,set,set) = IFEQ($1,$2,IFEQ($2,$3,[id $1,id $1].-->
    id $1,{}),{});
    deffunc F(Element of A,set) = IFEQ($1,$2,{ id $1 },{});
    consider M being ManySortedSet of [:A,A:] such that
A1: for i,j being Element of A holds M.(i,j) = F(i,j) from MSSLambda2D;
    consider c being ManySortedSet of [:A,A,A:] such that
A2: for i,j,k being Element of A holds c.(i,j,k) = F(i,j,k) from
    MSSLambda3D;
A3: now
      let i;
      assume i in [:A,A,A:];
      then consider i1,i2,i3 being object such that
A4:   i1 in A & i2 in A & i3 in A and
A5:   i = [i1,i2,i3] by MCART_1:68;
      reconsider i1,i2,i3 as Element of A by A4;
      per cases;
      suppose that
A6:     i1 = i2 and
A7:     i2 = i3;
A8:     M.(i1,i1) = IFEQ(i1,i1,{ id i1 },{}) by A1
          .= {id i1} by FUNCOP_1:def 8;
A9:     c.i = c.(i1,i2,i3) by A5,MULTOP_1:def 1
          .= IFEQ(i1,i2,IFEQ(i2,i3,[id i1,id i1].-->id i1,{}),{}) by A2
          .= IFEQ(i2,i3,[id i1,id i1].-->id i1,{}) by A6,FUNCOP_1:def 8
          .= [id i1,id i1].-->id i1 by A7,FUNCOP_1:def 8;
A10:    {|M|}.i = {|M|}.(i1,i1,i1) by A5,A6,A7,MULTOP_1:def 1
          .= {id i1} by A8,Def3;
A11:    {|M,M|}.i = {|M,M|}.(i1,i1,i1) by A5,A6,A7,MULTOP_1:def 1
          .= [:{id i1},{id i1}:] by A8,Def4
          .= {[id i1,id i1]} by ZFMISC_1:29
          .= dom([id i1,id i1].-->id i1);
        thus c.i is Function of {|M,M|}.i, {|M|}.i
        by A9,A10,A11;
      end;
      suppose
A12:    i1 <> i2 or i2 <> i3;
A13:    now
          per cases by A12;
          suppose
A14:        i1 <> i2;
            thus c.i = c.(i1,i2,i3) by A5,MULTOP_1:def 1
              .= IFEQ(i1,i2,IFEQ(i2,i3,[id i1,id i1].-->id i1,{}),{}) by A2
              .= {} by A14,FUNCOP_1:def 8;
          end;
          suppose that
A15:        i1 = i2 and
A16:        i2 <> i3;
            thus c.i = c.(i1,i2,i3) by A5,MULTOP_1:def 1
              .= IFEQ(i1,i2,IFEQ(i2,i3,[id i1,id i1].-->id i1,{}),{}) by A2
              .= IFEQ(i2,i3,[id i1,id i1].-->id i1,{}) by A15,FUNCOP_1:def 8
              .= {} by A16,FUNCOP_1:def 8;
          end;
        end;
        M.(i1,i2) = IFEQ(i1,i2,{ id i1 },{}) & M.(i2,i3) = IFEQ(i2,i3,{
        id i2 },{}) by A1;
        then
A17:    M.(i1,i2) = {} or M.(i2,i3) = {} by A12,FUNCOP_1:def 8;
        {|M,M|}.i = {|M,M|}.(i1,i2,i3) by A5,MULTOP_1:def 1
          .= [:M.(i2,i3),M.(i1,i2):] by Def4
          .= {} by A17;
        hence c.i is Function of {|M,M|}.i, {|M|}.i
        by A13,FUNCT_2:2,RELAT_1:38,XBOOLE_1:2;
      end;
    end;
    c is Function-yielding
    proof
      let i be object;
      assume i in dom c;
      then i in [:A,A,A:];
      hence thesis by A3;
    end;
    then reconsider c as ManySortedFunction of [:A,A,A:];
    reconsider c as BinComp of M by A3,PBOOLE:def 15;
    set C = AltCatStr(#A,M,c#);
    C is quasi-discrete
    proof
      let o1,o2 be Object of C;
      assume that
A18:  <^o1,o2^> <> {} and
A19:  o1 <> o2;
      <^o1,o2^> = IFEQ(o1,o2,{ id o1 },{}) by A1
        .= {} by A19,FUNCOP_1:def 8;
      hence contradiction by A18;
    end;
    then reconsider C = AltCatStr(#A,M,c#) as quasi-discrete strict non empty
    AltCatStr;
    take C;
    thus the carrier of C = A;
    let i be Object of C;
    thus <^i,i^> = IFEQ(i,i,{ id i },{}) by A1
      .= { id i } by FUNCOP_1:def 8;
  end;
  correctness
  proof
    let C1,C2 be quasi-discrete strict non empty AltCatStr such that
A20: the carrier of C1 = A and
A21: for i being Object of C1 holds <^i,i^> = { id i } and
A22: the carrier of C2 = A and
A23: for i being Object of C2 holds <^i,i^> = { id i };
A24: now
      let i,j,k;
      assume that
A25:  i in A and
A26:  j in A & k in A;
      reconsider i2 = i as Object of C2 by A22,A25;
      reconsider i1 = i as Object of C1 by A20,A25;
      per cases;
      suppose
A27:    i = j & j = k;
A28:    <^i2,i2^> = { id i2 } & (the Comp of C2).(i2,i2,i2) is Function
        of [:<^i2,i2 ^>,<^i2,i2^>:],<^i2,i2^> by A23;
         reconsider ii=i as set by TARSKI:1;
        <^i1,i1^> = { id i1 } & (the Comp of C1).(i1,i1,i1) is Function
        of [:<^i1,i1 ^>,<^i1,i1^>:],<^i1,i1^> by A21;
        hence (the Comp of C1).(i,j,k) = (id ii,id ii) :->id ii by A27,
FUNCOP_1:def 10
          .= (the Comp of C2).(i,j,k) by A27,A28,FUNCOP_1:def 10;
      end;
      suppose
A29:    i <> j or j <> k;
        reconsider j1 = j, k1 = k as Object of C1 by A20,A26;
A30:    <^i1,j1^> = {} or <^j1,k1^> = {} by A29,Def18;
        reconsider j2 = j, k2 = k as Object of C2 by A22,A26;
A31:    (the Comp of C2).(i2,j2,k2) is Function of [:<^j2,k2^>,<^i2,j2^>
:],<^i2,k2^> & (the Comp of C1).(i1,j1,k1) is Function of [:<^j1,k1^>,<^i1,j1^>
        :],<^i1,k1^>;
        <^i2,j2^> = {} or <^j2,k2^> = {} by A29,Def18;
        hence (the Comp of C1).(i,j,k) = (the Comp of C2).(i,j,k) by A30,A31;
      end;
    end;
    now
      let i,j be Element of A;
      reconsider i2 = i as Object of C2 by A22;
      reconsider i1 = i as Object of C1 by A20;
      per cases;
      suppose
A32:    i = j;
        hence (the Arrows of C1).(i,j) = <^i1,i1^> .= { id i } by A21
          .= <^i2,i2^> by A23
          .= (the Arrows of C2).(i,j) by A32;
      end;
      suppose
A33:    i <> j;
        reconsider j2 = j as Object of C2 by A22;
        reconsider j1 = j as Object of C1 by A20;
        thus (the Arrows of C1).(i,j) = <^i1,j1^> .= {} by A33,Def18
          .= <^i2,j2^> by A33,Def18
          .= (the Arrows of C2).(i,j);
      end;
    end;
    then the Arrows of C1 = the Arrows of C2 by A20,A22,Th3;
    hence thesis by A20,A22,A24,Th4;
  end;
end;

registration
  cluster quasi-discrete -> transitive for AltCatStr;
  coherence;
end;

theorem Th18:
  for A being non empty set, o1,o2,o3 being Object of DiscrCat A
  st o1 <> o2 or o2 <> o3 holds (the Comp of DiscrCat A).(o1,o2,o3) = {}
proof
  let A be non empty set, o1,o2,o3 be Object of DiscrCat A;
  assume o1 <> o2 or o2 <> o3;
  then <^o1,o2^> = {} or <^o2,o3^> = {} by Def18;
  hence thesis;
end;

theorem Th19:
  for A being non empty set, o being Object of DiscrCat A holds (
  the Comp of DiscrCat A).(o,o,o) = (id o,id o) :-> id o
proof
  let A be non empty set, o be Object of DiscrCat A;
  <^o,o^> = {id o} by Def20;
  hence thesis by FUNCOP_1:def 10;
end;

registration
  let A be non empty set;
  cluster DiscrCat A -> pseudo-functional pseudo-discrete with_units
    associative;
  coherence
  proof
    set C = DiscrCat A;
    thus C is pseudo-functional
    proof
      let o1,o2,o3 be Object of C;
A1:   id o1 in Funcs(o1,o1) by FUNCT_2:126;
      per cases;
      suppose
A2:     o1 = o2 & o2 = o3;
        then
A3:     <^o2,o3^> = {id o1} by Def20;
        then
A4:     <^o1,o2^> c= Funcs(o1,o2) by A1,A2,ZFMISC_1:31;
        thus (the Comp of C).(o1,o2,o3) = (id o1,id o1) :-> id o1 by A2,Th19
          .= FuncComp({id o1},{id o1}) by Th7
          .= FuncComp(Funcs(o1,o2),Funcs(o2,o3))|([:<^o2,o3^>,<^o1,o2^>:]
        qua set) by A2,A3,A4,Th8;
      end;
      suppose
A5:     o1 <> o2 or o2 <> o3;
        then
A6:     <^o2,o3^> = {} or <^o1,o2^> = {} by Def18;
        thus (the Comp of C).(o1,o2,o3) = {} by A5,Th18
          .= FuncComp(Funcs(o1,o2),Funcs(o2,o3))|([:<^o2,o3^>,<^o1,o2^>:]
        qua set) by A6;
      end;
    end;
    thus C is pseudo-discrete
    proof
      let i be Object of C;
      <^i,i^> = { id i } by Def20;
      hence thesis;
    end;
    thus C is with_units
    proof
      thus the Comp of C is with_left_units
      proof
        let j be Element of C;
        reconsider j9=j as Object of C;
        take id j9;
        (the Arrows of C).(j,j) = <^j9,j9^> .= { id j9 } by Def20;
        hence id j9 in (the Arrows of C).(j,j) by TARSKI:def 1;
        let i be Element of C, f be set such that
A7:     f in (the Arrows of C).(i,j);
        reconsider i9=i as Object of C;
A8:     (the Arrows of C).(i,j) = <^i9,j9^>;
        then
A9:     i9 = j9 by A7,Def18;
        then f in { id i9} by A7,A8,Def20;
        then
A10:    f = id i9 by TARSKI:def 1;
        thus (the Comp of C).(i,j,j).(id j9,f) = ((id i9,id i9):->id i9).(id
        j9,f) by A9,Th19
          .= f by A9,A10,FUNCT_4:80;
      end;
      let j be Element of C;
      reconsider j9=j as Object of C;
      take id j9;
      (the Arrows of C).(j,j) = <^j9,j9^> .= { id j9 } by Def20;
      hence id j9 in (the Arrows of C).(j,j) by TARSKI:def 1;
      let i be Element of C, f be set such that
A11:  f in (the Arrows of C).(j,i);
      reconsider i9=i as Object of C;
A12:  (the Arrows of C).(j,i) = <^j9,i9^>;
      then
A13:  i9 = j9 by A11,Def18;
      then f in { id i9} by A11,A12,Def20;
      then
A14:  f = id i9 by TARSKI:def 1;
      thus (the Comp of C).(j,j,i).(f,id j9) = ((id i9,id i9):->id i9).(f,id
      j9) by A13,Th19
        .= f by A13,A14,FUNCT_4:80;
    end;
    thus C is associative
    proof
      let i,j,k,l be Element of C;
      set G = the Arrows of C, c = the Comp of C;
      reconsider i9=i, j9=j, k9=k, l9=l as Object of C;
      let f,g,h be set;
      assume that
A15:  f in G.(i,j) and
A16:  g in G.(j,k) and
A17:  h in G.(k,l);
      f in <^i9,j9^> by A15;
      then
A18:  i9 = j9 by Def18;
A19:  <^i9,i9^> = { id i9 } by Def20;
      then
A20:  f = id i9 by A15,A18,TARSKI:def 1;
      g in <^j9,k9^> by A16;
      then
A21:  j9 = k9 by Def18;
      then
A22:  g = id i9 by A16,A18,A19,TARSKI:def 1;
A23:  c.(i9,i9,i9) = (id i9,id i9) :-> id i9 by Th19;
      h in <^k9,l9^> by A17;
      then
A24:  k9 = l9 by Def18;
      then h = id i9 by A17,A18,A21,A19,TARSKI:def 1;
      hence thesis by A18,A21,A24,A20,A22,A23,FUNCT_4:80;
    end;
  end;
end;