formal/mizar/altcat_5.miz

:: Products in Categories without Uniqueness of { \bf cod } and { \bf dom
:: }
:: http://creativecommons.org/licenses/by-sa/3.0/.

environ

 vocabularies ALTCAT_1, CAT_1, RELAT_1, ALTCAT_3, CAT_3, FUNCT_1, PBOOLE,
      ALTCAT_5, FUNCOP_1, CARD_1, FUNCT_2, XBOOLE_0, SUBSET_1, STRUCT_0,
      TARSKI, PARTFUN1, CARD_3, MSUALG_6;
 notations TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1,
      FUNCT_2, PBOOLE, CARD_3, FUNCOP_1, NUMBERS, STRUCT_0, ALTCAT_1, ALTCAT_3;
 constructors ALTCAT_3, RELSET_1, CARD_3, NUMBERS;
 registrations XBOOLE_0, RELSET_1, FUNCOP_1, STRUCT_0, ALTCAT_1, FUNCT_2,
      FUNCT_1, CARD_3, RELAT_1;
 requirements SUBSET, BOOLE;
 definitions TARSKI, RELAT_1, FUNCOP_1, ALTCAT_3;
 equalities ALTCAT_1, ORDINAL1;
 expansions PARTFUN1;
 theorems FUNCT_2, FUNCOP_1, CARD_1, TARSKI, ALTCAT_1, FUNCT_5, FUNCT_1,
      ALTCAT_3, PARTFUN1, YELLOW17, RELAT_1, CARD_3, PBOOLE;
 schemes PBOOLE, CLASSES1;

begin

reserve
  I for set,
  E for non empty set;

registration
  cluster empty -> {}-defined for Relation;
  coherence
  proof
    let R be Relation;
    assume R is empty;
    hence dom R c= {};
  end;
end;

definition
  let C be AltGraph;
  attr C is functional means
:Def1:
  for a, b being Object of C holds <^a,b^> is functional;
end;

registration
  let E;
  cluster EnsCat E -> functional;
  coherence
  proof
    let a, b be Object of EnsCat E;
    <^a,b^> = Funcs(a,b) by ALTCAT_1:def 14;
    hence thesis;
  end;
end;

registration
  cluster functional strict for category;
  existence
  proof
    take EnsCat {{}};
    thus thesis;
  end;
end;

registration
  let C be functional AltCatStr;
  cluster the AltGraph of C -> functional;
  coherence
  proof
    let a,b be Object of the AltGraph of C;
    reconsider a1 = a, b1 = b as Object of C;
    <^a1,b1^> is functional by Def1;
    hence thesis;
  end;
end;

registration
  cluster functional strict for AltGraph;
  existence
  proof
    take the AltGraph of EnsCat {{}};
    thus thesis;
  end;
end;

registration
  cluster functional strict for category;
  existence
  proof
    take EnsCat {{}};
    thus thesis;
  end;
end;

registration
  let C be functional AltGraph;
  let a,b be Object of C;
  cluster <^a,b^> -> functional;
  coherence by Def1;
end;

reconsider a = 0, b = 1 as Element of 2 by CARD_1:50,TARSKI:def 2;

set C = EnsCat {{}};

Lm1: the carrier of C = {0} by ALTCAT_1:def 14;

reconsider o = {} as Object of C by Lm1,TARSKI:def 1;

Lm2: Funcs({} qua set,{} qua set) = {{}} by FUNCT_5:57;

Lm3:
now
  let o1,o be Object of C;
A1: o1 = {} & o = {} by Lm1,TARSKI:def 1;
  <^o1,o^> = Funcs(o1,o) by ALTCAT_1:def 14;
  hence {} is Morphism of o1,o & {} in <^o1,o^> by A1,Lm1,Lm2;
end;

Lm4:
now
  let o1, o be Object of C;
  let m1 be Morphism of o1,o;
A1: o = {} & o1 = {} by Lm1,TARSKI:def 1;
  <^o1,o^> = Funcs(o1,o) by ALTCAT_1:def 14;
  hence m1 = {} by A1,Lm2,TARSKI:def 1;
end;

Lm5:
now
  let o1,o be Object of C;
  o = {} & o1 = {} by Lm1,TARSKI:def 1;
  hence o1 = o;
end;

Lm6:
now
  let o1,o be Object of C;
  let m1,m be Morphism of o1,o;
  thus m1 = {} by Lm4
  .= m by Lm4;
end;

definition
  let C be non empty AltCatStr;
  let I be set;
  mode ObjectsFamily of I,C is Function of I,C;
end;

definition
  let C be non empty AltCatStr;
  let o be Object of C;
  let I be set;
  let f be ObjectsFamily of I,C;
  mode MorphismsFamily of o,f -> ManySortedSet of I means
:Def2:
  for i being object st i in I
   ex o1 being Object of C st o1 = f.i & it.i is Morphism of o,o1;
  existence
  proof
    defpred P[object,object] means ex o1 being Object of C st o1 = f.$1 &
    $2 is Morphism of o,o1;
A1: for i being object st i in I ex j being object st P[i,j]
    proof
      let i be object;
      assume i in I;
      then reconsider o1 = f.i as Object of C by FUNCT_2:5;
      take the Morphism of o,o1;
      thus thesis;
    end;
    ex f being ManySortedSet of I st
    for i being object st i in I holds P[i,f.i] from PBOOLE:sch 3(A1);
    hence thesis;
  end;
end;

definition
  let C be non empty AltCatStr;
  let o be Object of C;
  let I be non empty set;
  let f be ObjectsFamily of I,C;
  redefine mode MorphismsFamily of o,f means
:Def3:
  for i being Element of I holds it.i is Morphism of o,f.i;
  compatibility
  proof
    let F be ManySortedSet of I;
    hereby
      assume
A1:   F is MorphismsFamily of o,f;
      let i be Element of I;
      ex o1 being Object of C st o1 = f.i & F.i is Morphism of o,o1
      by A1,Def2;
      hence F.i is Morphism of o,f.i;
    end;
    assume
A2: for i being Element of I holds F.i is Morphism of o,f.i;
    let i be object;
    assume i in I;
    then reconsider j = i as Element of I;
    take f.j;
    thus thesis by A2;
  end;
end;

definition
  let C be non empty AltCatStr;
  let o be Object of C;
  let I be non empty set;
  let f be ObjectsFamily of I,C;
  let M be MorphismsFamily of o,f;
  let i be Element of I;
  redefine func M.i -> Morphism of o,f.i;
  coherence by Def3;
end;

registration
  let C be functional non empty AltCatStr;
  let o be Object of C;
  let I be set;
  let f be ObjectsFamily of I,C;
  cluster -> Function-yielding for MorphismsFamily of o,f;
  coherence
  proof
    let F be MorphismsFamily of o,f;
    let i be object;
    assume i in dom F;
    then ex o1 being Object of C st
    o1 = f.i & F.i is Morphism of o,o1 by Def2;
    hence thesis;
  end;
end;

theorem Th1:
  for C being non empty AltCatStr, o being Object of C
  for f being ObjectsFamily of {},C holds
  {} is MorphismsFamily of o,f
  proof
    let C be non empty AltCatStr, o be Object of C, f be ObjectsFamily of {},C;
    reconsider A = {} as {}-defined Relation;
    A is total;
    then reconsider A = {} as ManySortedSet of {};
    A is MorphismsFamily of o,f
    proof
      let i be object;
      thus thesis;
    end;
    hence thesis;
 end;

definition
  let C be non empty AltCatStr;
  let I be set;
  let A be ObjectsFamily of I,C;
  let B be Object of C;
  let P be MorphismsFamily of B,A;
  attr P is feasible means
  for i being set st i in I ex o being Object of C st o = A.i & P.i in <^B,o^>;
end;

definition
  let C be non empty AltCatStr;
  let I be non empty set;
  let A be ObjectsFamily of I,C;
  let B be Object of C;
  let P be MorphismsFamily of B,A;
  redefine attr P is feasible means :Def5:
  for i being Element of I holds P.i in <^B,A.i^>;
  compatibility
  proof
    thus P is feasible implies
    for i being Element of I holds P.i in <^B,A.i^>
    proof
      assume
A1:   P is feasible;
      let i be Element of I;
      ex o being Object of C st o = A.i & P.i in <^B,o^> by A1;
      hence thesis;
    end;
    assume
A2: for i being Element of I holds P.i in <^B,A.i^>;
    let i be set;
    assume i in I;
    then reconsider i as Element of I;
    reconsider A as ObjectsFamily of I,C;
    take A.i;
    thus thesis by A2;
  end;
end;

definition
  let C be category;
  let I be set;
  let A be ObjectsFamily of I,C;
  let B be Object of C; :: product object
  let P be MorphismsFamily of B,A; :: product family
  attr P is projection-morphisms means
  for X being Object of C, F being MorphismsFamily of X,A
  st F is feasible
  ex f being Morphism of X,B st f in <^X,B^> &
::existence
  (for i being set st i in I
  ex si being Object of C, Pi being Morphism of B,si st
  si = A.i & Pi = P.i & F.i = Pi * f) &
::uniqueness
  for f1 being Morphism of X,B st for i being set st i in I
  ex si being Object of C, Pi being Morphism of B,si st
  si = A.i & Pi = P.i & F.i = Pi * f1
  holds f = f1;
end;

definition
  let C be category;
  let I be non empty set;
  let A be ObjectsFamily of I,C;
  let B be Object of C;
  let P be MorphismsFamily of B,A;
  redefine attr P is projection-morphisms means

  for X being Object of C, F being MorphismsFamily of X,A st F is feasible
   ex f being Morphism of X,B st f in <^X,B^> &
::existence
    (for i being Element of I holds F.i = P.i * f) &
::uniqueness
    for f1 being Morphism of X,B st
      for i being Element of I holds F.i = P.i * f1
    holds f = f1;
  correctness
  proof
    thus P is projection-morphisms implies
    for Y being Object of C, F being MorphismsFamily of Y,A st F is feasible
    ex f being Morphism of Y,B st f in <^Y,B^> &
    (for i being Element of I holds F.i = P.i * f) &
    for f1 being Morphism of Y,B st
    for i being Element of I holds F.i = P.i * f1
    holds f = f1
    proof
      assume
A1:   P is projection-morphisms;
      let Y be Object of C, F be MorphismsFamily of Y,A;
      assume
A2:   F is feasible;
      consider f being Morphism of Y,B such that
A3:   f in <^Y,B^> and
A4:   for i being set st i in I
      ex si being Object of C, Pi being Morphism of B,si st
      si = A.i & Pi = P.i & F.i = Pi * f and
A5:   for f1 being Morphism of Y,B st for i being set st i in I
      ex si being Object of C, Pi being Morphism of B,si st
      si = A.i & Pi = P.i & F.i = Pi * f1
      holds f = f1 by A2,A1;
      take f;
      thus f in <^Y,B^> by A3;
      hereby
        let i be Element of I;
        ex si being Object of C, Pi being Morphism of B,si st
        si = A.i & Pi = P.i & F.i = Pi * f by A4;
        hence F.i = P.i * f;
      end;
      let f1 be Morphism of Y,B such that
A6:   for i being Element of I holds F.i = P.i * f1;
      for i being set st i in I
      ex si being Object of C, Pi being Morphism of B,si st
      si = A.i & Pi = P.i & F.i = Pi * f1
      proof
        let i be set;
        assume i in I;
        then reconsider i as Element of I;
        reconsider si = A.i as Object of C;
        reconsider Pi = P.i as Morphism of B,si;
        take si, Pi;
        thus thesis by A6;
      end;
      hence thesis by A5;
    end;
    assume
A7: for Y being Object of C, F being MorphismsFamily of Y,A st F is feasible
    ex f being Morphism of Y,B st f in <^Y,B^> &
    (for i being Element of I holds F.i = P.i * f) &
    for f1 being Morphism of Y,B st
    for i being Element of I holds F.i = P.i * f1
    holds f = f1;
    let Y be Object of C, F be MorphismsFamily of Y,A;
    assume F is feasible;
    then consider f be Morphism of Y,B such that
A8: f in <^Y,B^> and
A9: for i being Element of I holds F.i = P.i * f and
A10: for f1 being Morphism of Y,B st
    for i being Element of I holds F.i = P.i * f1
    holds f = f1 by A7;
    take f;
    thus f in <^Y,B^>  by A8;
    thus for i being set st i in I
    ex si being Object of C, Pi being Morphism of B,si st
    si = A.i & Pi = P.i & F.i = Pi * f
    proof
      let i be set;
      assume i in I;
      then reconsider j = i as Element of I;
      take A.j, P.j;
      thus thesis by A9;
    end;
    let f1 be Morphism of Y,B such that
A11: for i being set st i in I
    ex si being Object of C, Pi being Morphism of B,si st
    si = A.i & Pi = P.i & F.i = Pi * f1;
    now
      let i be Element of I;
      ex si being Object of C, Pi being Morphism of B,si st
      si = A.i & Pi = P.i & F.i = Pi * f1 by A11;
      hence F.i = P.i * f1;
    end;
    hence thesis by A10;
  end;
end;

registration
  let C be category, A be ObjectsFamily of {},C;
  let B be Object of C;
  cluster -> feasible for MorphismsFamily of B,A;
  coherence;
end;

theorem Th2:
  for C being category, A being ObjectsFamily of {},C
  for B being Object of C st B is terminal holds
  ex P being MorphismsFamily of B,A st P is empty projection-morphisms
  proof
    let C be category;
    let A be ObjectsFamily of {},C;
    let B be Object of C;
    assume
A1: B is terminal;
    reconsider P = {} as MorphismsFamily of B,A by Th1;
    take P;
    thus P is empty;
    let X be Object of C, F be MorphismsFamily of X,A;
    assume F is feasible;
    consider f being Morphism of X,B such that
A2: f in <^X,B^> &
    for M1 being Morphism of X,B st M1 in <^X,B^> holds f = M1
    by A1,ALTCAT_3:27;
    take f;
    thus thesis by A2;
  end;

theorem Th3:
  for A being ObjectsFamily of I,EnsCat {{}}, o being Object of EnsCat {{}}
  holds I --> {} is MorphismsFamily of o,A
  proof
    let A be ObjectsFamily of I,C;
    let o be Object of C;
    let i be object such that
A1: i in I;
    reconsider I as non empty set by A1;
    reconsider j = i as Element of I by A1;
    reconsider A1 = A as ObjectsFamily of I,C;
    reconsider o1 = A1.j as Object of C;
    take o1;
    thus o1 = A.i;
    thus thesis by Lm3;
  end;

theorem Th4:
  for A being ObjectsFamily of I,EnsCat {{}},
      o being Object of EnsCat {{}},
      P being MorphismsFamily of o,A st P = I --> {} holds
   P is feasible projection-morphisms
   proof
     let A be ObjectsFamily of I,EnsCat {{}};
     let o be Object of EnsCat {{}};
     let P be MorphismsFamily of o,A;
     assume
A1:  P = I --> {};
     thus P is feasible
     proof
       let i be set;
       assume
A2:    i in I;
       then reconsider I as non empty set;
       reconsider i as Element of I by A2;
       reconsider A as ObjectsFamily of I,C;
       P.i = {} by A1;
       then P.i in <^o,A.i^> by Lm3;
       hence thesis;
     end;
     let Y be Object of C, F being MorphismsFamily of Y,A;
     assume F is feasible;
     reconsider f = {} as Morphism of Y,o by Lm3;
     take f;
     thus f in <^Y,o^> by Lm3;
     thus for i being set st i in I
     ex si being Object of C, Pi being Morphism of o,si st
     si = A.i & Pi = P.i & F.i = Pi * f
     proof
       let i be set;
       assume
A3:    i in I;
       then reconsider I as non empty set;
       reconsider j = i as Element of I by A3;
       reconsider M = {} as Morphism of o,o by Lm3;
       reconsider A1 = A as ObjectsFamily of I,C;
       reconsider F1 = F as MorphismsFamily of Y,A1;
       take o, M;
       A1.j = {} by Lm1,TARSKI:def 1;
       hence o = A.i by Lm5;
       thus M = P.i by A1;
       F1.j is Morphism of Y,o & M*f is Morphism of Y,o by Lm5;
       hence thesis by Lm6;
     end;
     thus thesis by Lm4;
   end;

definition
  let C be category;
  attr C is with_products means
:Def8:
  for I being set, A being ObjectsFamily of I,C
   ex B being Object of C, P being MorphismsFamily of B,A st
    P is feasible projection-morphisms;
end;

registration
  cluster EnsCat {{}} -> with_products;
  coherence
  proof
    let I be set, A be ObjectsFamily of I,C;
    reconsider P = I --> {} as MorphismsFamily of o,A by Th3;
    take o,P;
    thus thesis by Th4;
  end;
end;

registration
  cluster with_products for category;
  existence
  proof
    take EnsCat {{}};
    thus thesis;
  end;
end;

definition
  let C be category;
  let I be set, A be ObjectsFamily of I,C;
  let B be Object of C;
  attr B is A-CatProduct-like means

  ex P being MorphismsFamily of B,A st P is feasible projection-morphisms;
end;

registration
  let C be with_products category;
  let I be set, A be ObjectsFamily of I,C;
  cluster A-CatProduct-like for Object of C;
  existence
  proof
    consider B being Object of C, P being MorphismsFamily of B,A such that
A1: P is feasible projection-morphisms by Def8;
    take B,P;
    thus thesis by A1;
  end;
end;

registration
  let C be category;
  let A be ObjectsFamily of {},C;
  cluster A-CatProduct-like -> terminal for Object of C;
  coherence
  proof
    let B be Object of C such that
A1: B is A-CatProduct-like;
    for X being Object of C
    ex M being Morphism of X,B st M in <^X,B^> &
    for M1 being Morphism of X,B st M1 in <^X,B^> holds M = M1
    proof
      let X be Object of C;
      consider P being MorphismsFamily of B,A such that
A2:   P is feasible projection-morphisms by A1;
      set F = the MorphismsFamily of X,A;
      consider f being Morphism of X,B such that
A3:   f in <^X,B^> and
      for i being set st i in {}
      ex si being Object of C, Pi being Morphism of B,si st
      si = A.i & Pi = P.i & F.i = Pi * f and
A4:   for f1 being Morphism of X,B st for i being set st i in {}
      ex si being Object of C, Pi being Morphism of B,si st
      si = A.i & Pi = P.i & F.i = Pi * f1
      holds f = f1 by A2;
      take f;
      thus f in <^X,B^> by A3;
      let M be Morphism of X,B;
      for i being set st i in {}
      ex si being Object of C, Pi being Morphism of B,si st
      si = A.i & Pi = P.i & F.i = Pi * M;
      hence thesis by A4;
    end;
    hence thesis by ALTCAT_3:27;
  end;
end;

theorem
  for C being category, A being ObjectsFamily of {},C
  for B being Object of C st B is terminal holds
  B is A-CatProduct-like
  proof
    let C be category;
    let A be ObjectsFamily of {},C;
    let B be Object of C;
    assume B is terminal;
    then ex P being MorphismsFamily of B,A st
    P is empty projection-morphisms by Th2;
    hence thesis;
  end;

theorem
  for C being category, A being ObjectsFamily of I,C,
      C1,C2 being Object of C
  st C1 is A-CatProduct-like & C2 is A-CatProduct-like
  holds C1,C2 are_iso
  proof
    let C be category;
    let A be ObjectsFamily of I,C;
    let C1,C2 be Object of C;
    assume that
A1: C1 is A-CatProduct-like and
A2: C2 is A-CatProduct-like;
    per cases;
    suppose I is empty;
      hence thesis by A1,A2,ALTCAT_3:28;
    end;
    suppose I is non empty;
      then reconsider I as non empty set;
      reconsider A as ObjectsFamily of I,C;
      consider P1 being MorphismsFamily of C1,A such that
A3:   P1 is feasible and
A4:   P1 is projection-morphisms by A1;
      consider P2 being MorphismsFamily of C2,A such that
A5:   P2 is feasible and
A6:   P2 is projection-morphisms by A2;
      consider f1 being Morphism of C2,C1 such that
A7:   f1 in <^C2,C1^> and
A8:   for i being Element of I holds P2.i = P1.i * f1 and
      for fa being Morphism of C2,C1 st
      for i being Element of I holds P2.i = P1.i * fa
      holds f1 = fa by A4,A5;
      consider g1 being Morphism of C1,C1 such that
      g1 in <^C1,C1^> and
      for i being Element of I holds P1.i = P1.i * g1 and
A9:   for fa being Morphism of C1,C1 st
      for i being Element of I holds P1.i = P1.i * fa
      holds g1 = fa by A3,A4;
      consider f2 being Morphism of C1,C2 such that
A10:   f2 in <^C1,C2^> and
A11:   for i being Element of I holds P1.i = P2.i * f2 and
      for fa being Morphism of C1,C2 st
      for i being Element of I holds P1.i = P2.i * fa
      holds f2 = fa by A3,A6;
      consider g2 being Morphism of C2,C2 such that
      g2 in <^C2,C2^> and
      for i being Element of I holds P2.i = P2.i * g2 and
A12:  for fa being Morphism of C2,C2 st
      for i being Element of I holds P2.i = P2.i * fa
      holds g2 = fa by A5,A6;
      thus <^C1,C2^> <> {} & <^C2,C1^> <> {} by A7,A10;
      take f2;
A13:  f2 is retraction
      proof
        take f1;
        now
          let i be Element of I;
          P2.i in <^C2,A.i^> by A5;
          hence P2.i = P2.i * idm C2 by ALTCAT_1:def 17;
        end;
        then
A14:    g2 = idm C2 by A12;
        now
          let i be Element of I;
          P2.i in <^C2,A.i^> by A5;
          hence P2.i * (f2 * f1) = P2.i * f2 * f1 by A7,A10,ALTCAT_1:21
          .= P1.i * f1 by A11
          .= P2.i by A8;
        end;
        hence f2 * f1 = idm C2 by A14,A12;
      end;
      f2 is coretraction
      proof
        take f1;
        now
          let i be Element of I;
          P1.i in <^C1,A.i^> by A3;
          hence P1.i = P1.i * idm C1 by ALTCAT_1:def 17;
        end;
        then
A15:    g1 = idm C1 by A9;
        now
          let i be Element of I;
          P1.i in <^C1,A.i^> by A3;
          hence P1.i * (f1 * f2) = P1.i * f1 * f2 by A7,A10,ALTCAT_1:21
          .= P2.i * f2 by A8
          .= P1.i by A11;
        end;
        hence f1 * f2 = idm C1 by A15,A9;
      end;
      hence thesis by A7,A10,A13,ALTCAT_3:6;
    end;
  end;

reserve A for ObjectsFamily of I,EnsCat E;

definition
  let I,E,A;
  assume
A1: product A in E;
  func EnsCatProductObj A -> Object of EnsCat E equals :Def10:
  product A;
  coherence by A1,ALTCAT_1:def 14;
end;

definition
  let I,E,A;
  assume
A1: product A in E;
  func EnsCatProduct A -> MorphismsFamily of EnsCatProductObj A,A
  means :Def11:
  for i being set st i in I holds it.i = proj(A,i);
  existence
  proof
    deffunc F(object) = proj(A,$1);
    consider P being ManySortedSet of I such that
A2: for i being object st i in I holds P.i = F(i) from PBOOLE:sch 4;
    set B = EnsCatProductObj A;
A3: B = product A by A1,Def10;
    P is MorphismsFamily of B,A
    proof
      let i be object such that
A4:   i in I;
      reconsider I as non empty set by A4;
      reconsider i as Element of I by A4;
      reconsider A as ObjectsFamily of I,EnsCat E;
      take A.i;
A5:   <^B,A.i^> = Funcs(B,A.i) by ALTCAT_1:def 14;
      dom A = I by PARTFUN1:def 2;
      then
A6:   rng proj(A,i) c= A.i by YELLOW17:3;
      dom proj(A,i) = B by A3,PARTFUN1:def 2;
      then proj(A,i) in Funcs(B,A.i) by A6,FUNCT_2:def 2;
      hence thesis by A2,A5;
    end;
    then reconsider P as MorphismsFamily of B,A;
    take P;
    thus thesis by A2;
  end;
  uniqueness
  proof
    let f,g be MorphismsFamily of EnsCatProductObj A,A such that
A7: for i being set st i in I holds f.i = proj(A,i) and
A8: for i being set st i in I holds g.i = proj(A,i);
    now
      let i be object;
      assume
A9:   i in I;
      hence f.i = proj(A,i) by A7
      .= g.i by A8,A9;
    end;
    hence thesis by PBOOLE:3;
  end;
end;

theorem Th7:
  product A in E & product A = {} implies EnsCatProduct A = I --> {}
  proof
    assume that
A1: product A in E and
A2: product A = {};
    now
      let i be object;
      assume
   i in I;
      hence (EnsCatProduct A).i = proj(A,i) by A1,Def11
      .= {} by A2
      .= (I --> {}).i;
    end;
    hence thesis by PBOOLE:3;
  end;

theorem Th8:
  product A in E implies EnsCatProduct A is feasible projection-morphisms
  proof
    set B = EnsCatProductObj A;
    set P = EnsCatProduct A;
    assume
A1: product A in E; then
A2: B = product A by Def10;
    per cases;
    suppose
A3: product A <> {};
A4: dom A = I by PARTFUN1:def 2;
A5: now
      let i be set;
      assume i in I;
      then A.i in rng A by A4,FUNCT_1:def 3;
      hence A.i <> {} by A3,CARD_3:26;
    end;
    thus P is feasible
    proof
      let i be set;
      assume
A6:   i in I;
      then reconsider I as non empty set;
      reconsider i as Element of I by A6;
      reconsider A as ObjectsFamily of I,EnsCat E;
      reconsider P as MorphismsFamily of B,A;
      take A.i;
A7:   <^B,A.i^> = Funcs(B,A.i) by ALTCAT_1:def 14;
      A.i <> {} by A5;
      then Funcs(B,A.i) <> {};
      then P.i in <^B,A.i^> by A7;
      hence thesis;
    end;
    let X be Object of EnsCat E, F be MorphismsFamily of X,A;
    assume F is feasible;
A8: <^X,B^> = Funcs(X,B) by ALTCAT_1:def 14;
    defpred P[object,object] means
    ex M being ManySortedSet of I st
    (for i being set st i in I holds M.i = F.i.$1) & $2 = M;
A9: for x being object st x in X ex u being object st P[x,u]
    proof
      let x be object;
      assume x in X;
      deffunc I(object) = F.$1.x;
      consider f being ManySortedSet of I such that
A10:  for i being object st i in I holds f.i = I(i) from PBOOLE:sch 4;
      take f,f;
      thus thesis by A10;
    end;
    consider ff being Function such that
A11: dom ff = X and
A12: for x being object st x in X holds P[x,ff.x] from CLASSES1:sch 1(A9);
A13: rng ff c= B
     proof
       let y be object;
       assume y in rng ff;
       then consider x being object such that
A14:   x in dom ff and
A15:   ff.x = y by FUNCT_1:def 3;
       consider M being ManySortedSet of I such that
A16:   for i being set st i in I holds M.i = F.i.x and
A17:   ff.x = M by A11,A12,A14;
A18:   dom M = I by PARTFUN1:def 2;
       now
         let i be object;
         assume
A19:     i in dom A;
         then reconsider I as non empty set;
         reconsider j = i as Element of I by A19;
         reconsider A1 = A as ObjectsFamily of I,EnsCat E;
         reconsider F1 = F as MorphismsFamily of X,A1;
A20:     <^X,A1.j^> = Funcs(X,A1.j) by ALTCAT_1:def 14;
         A1.j <> {} by A5;
         then
A21:     dom(F1.j) = X & rng(F1.j) c= A1.j by A20,FUNCT_2:92;
         then
A22:     F1.j.x in rng(F1.j) by A14,A11,FUNCT_1:def 3;
         M.j = F.j.x by A16;
         hence M.i in A.i by A22,A21;
       end;
       hence thesis by A2,A4,A15,A17,A18,CARD_3:9;
     end;
     then reconsider ff as Morphism of X,B by A8,A11,FUNCT_2:def 2;
     take ff;
     thus
A23: ff in <^X,B^> by A8,A13,A11,FUNCT_2:def 2;
     thus for i being set st i in I
     ex si being Object of EnsCat E, Pi being Morphism of B,si st
     si = A.i & Pi = P.i & F.i = Pi * ff
     proof
       let i be set;
       assume
A24:   i in I;
       then reconsider I as non empty set;
       reconsider j = i as Element of I by A24;
       reconsider A1 = A as ObjectsFamily of I,EnsCat E;
       reconsider P1 = P as MorphismsFamily of B,A1;
       reconsider F1 = F as MorphismsFamily of X,A1;
       take A1.j;
       take P1.j;
       thus A1.j = A.i & P1.j = P.i;
       reconsider p = P1.j as Function;
A25:   <^B,A1.j^> = Funcs(B,A1.j) by ALTCAT_1:def 14;
A26:   A1.j <> {} by A5;
       then <^X,A1.j^> <> {} by A25,A23,ALTCAT_1:def 2;
       then
A27:   P1.j * ff = p * ff by A23,A26,A25,ALTCAT_1:16;
A28:   <^X,A1.j^> = Funcs(X,A1.j) by ALTCAT_1:def 14;
       then
A29:   dom(P1.j*ff) = X by A26,FUNCT_2:92;
A30:   dom(F1.j) = X by A26,A28,FUNCT_2:92;
       now
         let x be object;
         assume
A31:     x in dom(F1.j);
         then consider M being ManySortedSet of I such that
A32:     for i being set st i in I holds M.i = F.i.x and
A33:     ff.x = M by A12,A30;
A34:     dom proj(A,j) = B by A2,CARD_3:def 16;
A35:     ff.x in rng ff by A11,A30,A31,FUNCT_1:def 3;
         thus (p*ff).x = p.(ff.x) by A11,A30,A31,FUNCT_1:13
         .= proj(A,j).(ff.x) by A1,Def11
         .= M.j by A33,A34,A35,A13,CARD_3:def 16
         .= F1.j.x by A32;
       end;
       hence F.i = P1.j * ff by A27,A29,A26,A28,FUNCT_2:92,FUNCT_1:2;
     end;
     let f1 be Morphism of X,B such that
A36: for i being set st i in I
     ex si being Object of EnsCat E, Pi being Morphism of B,si st
     si = A.i & Pi = P.i & F.i = Pi * f1;
A37: f1 is Function of X,B by A8,A23,FUNCT_2:66;
     then
A38: dom f1 = X by A3,A2,FUNCT_2:def 1;
A39: rng f1 c= B by A37,RELAT_1:def 19;
     now
       let x be object;
       assume
A40:   x in dom ff;
       then
A41:   f1.x in rng f1 by A11,A38,FUNCT_1:def 3;
       reconsider h = f1.x as Function by A2,A37;
       consider M being ManySortedSet of I such that
A42:   for i being set st i in I holds M.i = F.i.x and
A43:   ff.x = M by A11,A12,A40;
A44:   dom h = I by A2,A4,A41,A39,CARD_3:9;
       now
         let i be object;
         assume
A45:     i in dom M;
         then consider si being Object of EnsCat E, Pi being Morphism of B,si
         such that
A46:     si = A.i & Pi = P.i and
A47:     F.i = Pi * f1 by A36;
A48:     P.i = proj(A,i) by A1,A45,Def11;
A49:     dom proj(A,i) = B by A2,CARD_3:def 16;
A50:     <^B,si^> = Funcs(B,si) by ALTCAT_1:def 14;
A51:     si <> {} by A5,A45,A46;
         then
A52:     <^X,si^> <> {} by A50,A23,ALTCAT_1:def 2;
         thus M.i = (Pi*f1).x by A47,A42,A45
         .= (Pi qua Function*f1).x by A50,A23,A51,A52,ALTCAT_1:16
         .= Pi.h by A38,A11,A40,FUNCT_1:13
         .= h.i by A39,A41,A46,A48,A49,CARD_3:def 16;
       end;
       hence ff.x = f1.x by A44,A43,FUNCT_1:2,PARTFUN1:def 2;
     end;
     hence thesis by A11,A38,FUNCT_1:2;
   end;
   suppose
A53: product A = {};
    thus P is feasible
    proof
      let i be set such that
A54:   i in I;
      reconsider I as non empty set by A54;
      reconsider i as Element of I by A54;
      reconsider A as ObjectsFamily of I,EnsCat E;
      take A.i;
A55:   <^B,A.i^> = Funcs(B,A.i) by ALTCAT_1:def 14
      .= {{}} by A2,A53,FUNCT_5:57;
      P.i = (I-->{}).i by A1,A53,Th7
      .= {};
      hence thesis by A55,TARSKI:def 1;
    end;
    let X be Object of EnsCat E, F be MorphismsFamily of X,A;
    assume
A56: F is feasible;
A57: now
      assume
A58:  X <> {};
      {} in rng A by A53,CARD_3:26;
      then consider i being object such that
A59:  i in dom A and
A60:  A.i = {} by FUNCT_1:def 3;
      reconsider I as non empty set by A59;
      reconsider i as Element of I by A59;
      reconsider A as ObjectsFamily of I,EnsCat E;
      <^X,A.i^> = Funcs(X,A.i) by ALTCAT_1:def 14
      .= {} by A58,A60;
      hence contradiction by A56,Def5;
    end;
A61: <^X,B^> = Funcs(X,B) by ALTCAT_1:def 14
     .= {{}} by A57,FUNCT_5:57;
     then reconsider f = {} as Morphism of X,B by TARSKI:def 1;
     take f;
     thus f in <^X,B^> by A61;
     thus for i being set st i in I
     ex si being Object of EnsCat E, Pi being Morphism of B,si st
     si = A.i & Pi = P.i & F.i = Pi * f
     proof
       let i be set such that
A62:   i in I;
       reconsider J = I as non empty set by A62;
       reconsider j = i as Element of J by A62;
       reconsider A1 = A as ObjectsFamily of J,EnsCat E;
       reconsider P1 = P as MorphismsFamily of B,A1;
       reconsider si = A1.j as Object of EnsCat E;
       reconsider Pi = P1.j as Morphism of B,si;
       reconsider F1 = F as MorphismsFamily of X,A1;
       reconsider F2 = F1.j as Morphism of X,si;
       take si, Pi;
       thus si = A.i & Pi = P.i;
A63:   <^B,si^> = Funcs(B,si) by ALTCAT_1:def 14
       .= {{}} by A2,A53,FUNCT_5:57;
       then
A64:   <^X,si^> <> {} by A61,ALTCAT_1:def 2;
A65:   Funcs(X,si) = {{}} by A57,FUNCT_5:57;
A66:   <^X,si^> = Funcs(X,si) by ALTCAT_1:def 14;
       thus F.i = F2
       .= {} by A65,A66,TARSKI:def 1
       .= Pi qua Function * f
       .= Pi * f by A63,A61,A64,ALTCAT_1:16;
     end;
     let f1 be Morphism of X,B;
     thus thesis by A61,TARSKI:def 1;
   end;
 end;

theorem
  product A in E implies EnsCatProductObj A is A-CatProduct-like
  proof
    assume
A1: product A in E;
    take EnsCatProduct A;
    thus thesis by A1,Th8;
end;

theorem
  (for I,A holds product A in E) implies EnsCat E is with_products
  proof
    assume
A1: for I,A holds product A in E;
    let I,A;
    take EnsCatProductObj A, EnsCatProduct A;
    product A in E by A1;
    hence thesis by Th8;
  end;