formal/mizar/altcat_3.miz

:: Basic properties of objects and morphisms. In categories without
:: uniqueness of { \bf cod } and { \bf dom }
:: http://creativecommons.org/licenses/by-sa/3.0/.

environ

 vocabularies ALTCAT_1, XBOOLE_0, CAT_1, RELAT_1, CAT_3, BINOP_1, RELAT_2,
      FUNCT_1, FUNCOP_1, TARSKI, FUNCT_2, SUBSET_1, SETFAM_1, ZFMISC_1,
      ALTCAT_3;
 notations TARSKI, ZFMISC_1, XBOOLE_0, SUBSET_1, SETFAM_1, RELAT_1, FUNCT_1,
      FUNCT_2, FUNCOP_1, STRUCT_0, ALTCAT_1;
 constructors SETFAM_1, ALTCAT_1, RELSET_1;
 registrations XBOOLE_0, SETFAM_1, FUNCT_1, RELSET_1, ALTCAT_1, ZFMISC_1;
 requirements SUBSET, BOOLE;
 definitions TARSKI;
 expansions TARSKI;
 theorems FUNCOP_1, RELAT_1, FUNCT_2, ZFMISC_1, ALTCAT_1, TARSKI, FUNCT_1,
      XBOOLE_0, XBOOLE_1;
 schemes FUNCT_1;

begin

definition
  let C be with_units non empty AltCatStr, o1, o2 be Object of C, A be
  Morphism of o1,o2, B be Morphism of o2,o1;
  pred A is_left_inverse_of B means

  A * B = idm o2;
end;

notation
  let C be with_units non empty AltCatStr, o1, o2 be Object of C, A be
  Morphism of o1,o2, B be Morphism of o2,o1;
  synonym B is_right_inverse_of A for A is_left_inverse_of B;
end;

definition
  let C be with_units non empty AltCatStr, o1, o2 be Object of C, A be
  Morphism of o1,o2;
  attr A is retraction means

  ex B being Morphism of o2,o1 st B is_right_inverse_of A;
end;

definition
  let C be with_units non empty AltCatStr, o1, o2 be Object of C, A be
  Morphism of o1,o2;
  attr A is coretraction means

  ex B being Morphism of o2,o1 st B is_left_inverse_of A;
end;

theorem Th1:
  for C being with_units non empty AltCatStr, o being Object of C
  holds idm o is retraction & idm o is coretraction
proof
  let C be with_units non empty AltCatStr, o be Object of C;
  <^o,o^> <> {} by ALTCAT_1:19;
  then (idm o) * (idm o) = idm o by ALTCAT_1:def 17;
  then idm o is_left_inverse_of idm o;
  hence thesis;
end;

definition
  let C be category, o1, o2 be Object of C such that
A1: <^o1,o2^> <> {} and
A2: <^o2,o1^> <> {};
  let A be Morphism of o1,o2 such that
A3: A is retraction coretraction;
  func A" -> Morphism of o2,o1 means
  :Def4:
  it is_left_inverse_of A & it is_right_inverse_of A;
  existence
  proof
    consider B1 being Morphism of o2,o1 such that
A4: B1 is_right_inverse_of A by A3;
    take B1;
    consider B2 being Morphism of o2,o1 such that
A5: B2 is_left_inverse_of A by A3;
    B1 = idm o1 * B1 by A2,ALTCAT_1:20
      .= B2 * A * B1 by A5
      .= B2 * (A * B1) by A1,A2,ALTCAT_1:21
      .= B2 * idm o2 by A4
      .= B2 by A2,ALTCAT_1:def 17;
    hence thesis by A4,A5;
  end;
  uniqueness
  proof
    let M1,M2 be Morphism of o2,o1 such that
A6: M1 is_left_inverse_of A and
    M1 is_right_inverse_of A and
    M2 is_left_inverse_of A and
A7: M2 is_right_inverse_of A;
    thus M1 = M1 * idm o2 by A2,ALTCAT_1:def 17
      .= M1 * (A * M2) by A7
      .= M1 * A * M2 by A1,A2,ALTCAT_1:21
      .= idm o1 * M2 by A6
      .= M2 by A2,ALTCAT_1:20;
  end;
end;

theorem Th2:
  for C being category, o1,o2 being Object of C st <^o1,o2^> <> {}
  & <^o2,o1^> <> {} for A being Morphism of o1,o2 st A is retraction & A is
  coretraction holds A" * A = idm o1 & A * A" = idm o2
proof
  let C be category, o1,o2 be Object of C such that
A1: <^o1,o2^> <> {} & <^o2,o1^> <> {};
  let A be Morphism of o1,o2;
  assume A is retraction & A is coretraction;
  then A" is_left_inverse_of A & A" is_right_inverse_of A by A1,Def4;
  hence thesis;
end;

theorem Th3:
  for C being category, o1,o2 being Object of C st <^o1,o2^> <> {}
  & <^o2,o1^> <> {} for A being Morphism of o1,o2 st A is retraction & A is
  coretraction holds (A")" = A
proof
  let C be category, o1,o2 be Object of C such that
A1: <^o1,o2^> <> {} and
A2: <^o2,o1^> <> {};
  let A be Morphism of o1,o2;
  assume
A3: A is retraction & A is coretraction;
  then A" is_left_inverse_of A by A1,A2,Def4;
  then
A4: A" is retraction;
A5: A" is_right_inverse_of A by A1,A2,A3,Def4;
  then A" is coretraction;
  then
A6: (A")" is_right_inverse_of A" by A1,A2,A4,Def4;
  thus (A")" = idm o2 * ((A")") by A1,ALTCAT_1:20
    .= A * A" * (A")" by A5
    .= A * (A" * (A")") by A1,A2,ALTCAT_1:21
    .= A * idm o1 by A6
    .= A by A1,ALTCAT_1:def 17;
end;

theorem Th4:
  for C being category, o being Object of C holds (idm o)" = idm o
proof
  let C be category, o be Object of C;
A1: <^o,o^> <> {} by ALTCAT_1:19;
  idm o is retraction & idm o is coretraction by Th1;
  then
A2: (idm o)" is_left_inverse_of (idm o) by A1,Def4;
  thus (idm o)" = (idm o)" * idm o by A1,ALTCAT_1:def 17
    .= idm o by A2;
end;

definition
  let C be category, o1, o2 be Object of C, A be Morphism of o1,o2;
  attr A is iso means

  A*A" = idm o2 & A"*A = idm o1;
end;

theorem Th5:
  for C being category, o1, o2 being Object of C, A being Morphism
  of o1,o2 st A is iso holds A is retraction coretraction
proof
  let C be category, o1, o2 be Object of C, A be Morphism of o1,o2;
  assume
A1: A is iso;
  then A * A" = idm o2;
  then A" is_right_inverse_of A;
  hence A is retraction;
  A" * A = idm o1 by A1;
  then A" is_left_inverse_of A;
  hence thesis;
end;

theorem Th6:
  for C being category, o1,o2 being Object of C st <^o1,o2^> <> {}
  & <^o2,o1^> <> {} for A being Morphism of o1,o2 holds A is iso iff A is
  retraction & A is coretraction
proof
  let C be category, o1,o2 be Object of C such that
A1: <^o1,o2^> <> {} & <^o2,o1^> <> {};
  let A be Morphism of o1,o2;
  thus A is iso implies A is retraction & A is coretraction by Th5;
  assume
A2: A is retraction & A is coretraction;
  then A" is_right_inverse_of A by A1,Def4;
  then
A3: A * A" = idm o2;
  A" is_left_inverse_of A by A1,A2,Def4;
  then A" * A = idm o1;
  hence thesis by A3;
end;

theorem Th7:
  for C being category, o1,o2,o3 being Object of C, A being
Morphism of o1,o2, B being Morphism of o2,o3 st <^o1,o2^> <> {} & <^o2,o3^> <>
{} & <^o3,o1^> <> {} & A is iso & B is iso holds B * A is iso & (B * A)" = A" *
  B"
proof
  let C be category, o1,o2,o3 be Object of C, A be Morphism of o1,o2, B be
  Morphism of o2,o3;
  assume that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {} and
A3: <^o3,o1^> <> {};
  assume that
A4: A is iso and
A5: B is iso;
  consider A1 be Morphism of o2,o1 such that
A6: A1 = A";
A7: <^o2,o1^> <> {} by A2,A3,ALTCAT_1:def 2;
  then
A8: A is retraction & A is coretraction by A1,A4,Th6;
  consider B1 be Morphism of o3,o2 such that
A9: B1 = B";
A10: <^o3,o2^> <> {} by A1,A3,ALTCAT_1:def 2;
  then
A11: B is retraction & B is coretraction by A2,A5,Th6;
A12: (B*A)*(A1*B1) = B*(A*(A1*B1)) by A1,A2,A3,ALTCAT_1:21
    .= B*(A*A1*B1) by A1,A7,A10,ALTCAT_1:21
    .= B*((idm o2)*B1) by A1,A7,A8,A6,Th2
    .= B*B1 by A10,ALTCAT_1:20
    .= idm o3 by A2,A10,A11,A9,Th2;
  then
A13: (A1*B1) is_right_inverse_of (B*A);
  then
A14: (B*A) is retraction;
A15: <^o1,o3^> <> {} by A1,A2,ALTCAT_1:def 2;
  then
A16: (A1*B1)*(B*A) = A1*(B1*(B*A)) by A7,A10,ALTCAT_1:21
    .= A1*(B1*B*A) by A1,A2,A10,ALTCAT_1:21
    .= A1*((idm o2)*A) by A2,A10,A11,A9,Th2
    .= A1*A by A1,ALTCAT_1:20
    .= idm o1 by A1,A7,A8,A6,Th2;
  then
A17: (A1*B1) is_left_inverse_of (B*A);
  then (B*A) is coretraction;
  then A1*B1 = (B*A)" by A3,A15,A17,A13,A14,Def4;
  hence thesis by A6,A9,A16,A12;
end;

definition
  let C be category, o1, o2 be Object of C;
  pred o1,o2 are_iso means

  <^o1,o2^> <> {} & <^o2,o1^> <> {} & ex A being Morphism of o1,o2 st A is iso;
  reflexivity
  proof
    let o be Object of C;
    thus
A1: <^o,o^> <> {} & <^o,o^> <> {} by ALTCAT_1:19;
    take idm o;
    set A = idm o;
A2: A"*A = A * A by Th4
      .= idm o by A1,ALTCAT_1:def 17;
    A*A" = A * A by Th4
      .= idm o by A1,ALTCAT_1:def 17;
    hence thesis by A2;
  end;
  symmetry
  proof
    let o1,o2 be Object of C;
    assume that
A3: <^o1,o2^> <> {} & <^o2,o1^> <> {} and
A4: ex A being Morphism of o1,o2 st A is iso;
    thus <^o2,o1^> <> {} & <^o1,o2^> <> {} by A3;
    consider A being Morphism of o1,o2 such that
A5: A is iso by A4;
    take A1 = A";
A6: A is retraction & A is coretraction by A5,Th5;
    then
A7: A1"*A1 = A * A" by A3,Th3
      .= idm o2 by A3,A6,Th2;
    A1*A1" = A" * A by A3,A6,Th3
      .= idm o1 by A3,A6,Th2;
    hence thesis by A7;
  end;
end;

theorem
  for C being category, o1,o2,o3 being Object of C st o1,o2 are_iso & o2
  ,o3 are_iso holds o1,o3 are_iso
proof
  let C be category, o1,o2,o3 be Object of C such that
A1: o1,o2 are_iso and
A2: o2,o3 are_iso;
A3: <^o1,o2^> <> {} & <^o2,o3^> <> {} by A1,A2;
  consider B being Morphism of o2,o3 such that
A4: B is iso by A2;
  consider A being Morphism of o1,o2 such that
A5: A is iso by A1;
  <^o2,o1^> <> {} & <^o3,o2^> <> {} by A1,A2;
  hence
A6: <^o1,o3^> <> {} & <^o3,o1^> <> {} by A3,ALTCAT_1:def 2;
  take B * A;
  thus thesis by A3,A6,A5,A4,Th7;
end;

definition
  let C be non empty AltCatStr, o1, o2 be Object of C, A be Morphism of o1,o2;
  attr A is mono means

  for o being Object of C st <^o,o1^> <> {} for B,
  C being Morphism of o,o1 st A * B = A * C holds B = C;
end;

definition
  let C be non empty AltCatStr, o1, o2 be Object of C, A be Morphism of o1,o2;
  attr A is epi means

  for o being Object of C st <^o2,o^> <> {} for B,C
  being Morphism of o2,o st B * A = C * A holds B = C;
end;

theorem Th9:
  for C being associative transitive non empty AltCatStr, o1,o2,
o3 being Object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} for A being Morphism
  of o1,o2, B being Morphism of o2,o3 st A is mono & B is mono holds B * A is
  mono
proof
  let C be associative transitive non empty AltCatStr, o1,o2,o3 be Object of
  C;
  assume that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {};
  let A be Morphism of o1,o2, B be Morphism of o2,o3;
  assume that
A3: A is mono and
A4: B is mono;
  let o be Object of C;
  assume
A5: <^o,o1^> <> {};
  then
A6: <^o,o2^> <> {} by A1,ALTCAT_1:def 2;
  let M1,M2 be Morphism of o,o1;
  assume
A7: (B*A)*M1 = (B*A)*M2;
  (B*A)*M1 = B*(A*M1) & (B*A)*M2 = B*(A*M2) by A1,A2,A5,ALTCAT_1:21;
  then A*M1 = A*M2 by A4,A7,A6;
  hence thesis by A3,A5;
end;

theorem Th10:
  for C being associative transitive non empty AltCatStr, o1,o2,
o3 being Object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} for A being Morphism
  of o1,o2, B being Morphism of o2,o3 st A is epi & B is epi holds B * A is epi
proof
  let C be associative transitive non empty AltCatStr, o1,o2,o3 be Object of
  C;
  assume that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {};
  let A be Morphism of o1,o2, B be Morphism of o2,o3;
  assume that
A3: A is epi and
A4: B is epi;
  let o be Object of C;
  assume
A5: <^o3,o^> <> {};
  then
A6: <^o2,o^> <> {} by A2,ALTCAT_1:def 2;
  let M1,M2 be Morphism of o3,o;
  assume
A7: M1*(B*A) = M2*(B*A);
  M1*(B*A) = (M1*B)*A & M2*(B*A) = (M2*B)*A by A1,A2,A5,ALTCAT_1:21;
  then M1*B = M2*B by A3,A7,A6;
  hence thesis by A4,A5;
end;

theorem
  for C being associative transitive non empty AltCatStr, o1,o2,o3
being Object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} for A being Morphism of
  o1,o2, B being Morphism of o2,o3 st B * A is mono holds A is mono
proof
  let C be associative transitive non empty AltCatStr, o1,o2,o3 be Object of
  C;
  assume
A1: <^o1,o2^> <> {} & <^o2,o3^> <> {};
  let A be Morphism of o1,o2, B be Morphism of o2,o3;
  assume
A2: B * A is mono;
  let o be Object of C;
  assume
A3: <^o,o1^> <> {};
  let M1,M2 be Morphism of o,o1;
  assume
A4: A*M1 = A*M2;
  (B*A)*M1 = B*(A*M1) & (B*A)*M2 = B*(A*M2) by A1,A3,ALTCAT_1:21;
  hence thesis by A2,A3,A4;
end;

theorem
  for C being associative transitive non empty AltCatStr, o1,o2,o3
being Object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} for A being Morphism of
  o1,o2, B being Morphism of o2,o3 st B * A is epi holds B is epi
proof
  let C be associative transitive non empty AltCatStr, o1,o2,o3 be Object of
  C;
  assume
A1: <^o1,o2^> <> {} & <^o2,o3^> <> {};
  let A be Morphism of o1,o2, B be Morphism of o2,o3;
  assume
A2: B * A is epi;
  let o be Object of C;
  assume
A3: <^o3,o^> <> {};
  let M1,M2 be Morphism of o3,o;
  assume
A4: M1*B = M2*B;
  (M1*B)*A = M1*(B*A) & (M2*B)*A = M2*(B*A) by A1,A3,ALTCAT_1:21;
  hence thesis by A2,A3,A4;
end;

Lm1: now
  let C be with_units non empty AltCatStr, a be Object of C;
  thus idm a is epi
  proof
    let o be Object of C such that
A1: <^a,o^> <> {};
    let B, C be Morphism of a,o such that
A2: B * idm a = C * idm a;
    thus B = B * idm a by A1,ALTCAT_1:def 17
      .= C by A1,A2,ALTCAT_1:def 17;
  end;
  thus idm a is mono
  proof
    let o be Object of C such that
A3: <^o,a^> <> {};
    let B, C be Morphism of o,a such that
A4: idm a * B = idm a * C;
    thus B = idm a * B by A3,ALTCAT_1:20
      .= C by A3,A4,ALTCAT_1:20;
  end;
end;

theorem
  for X being non empty set for o1,o2 being Object of EnsCat X st <^o1,
  o2^> <> {} for A being Morphism of o1,o2, F being Function of o1,o2 st F = A
  holds A is mono iff F is one-to-one
proof
  let X be non empty set, o1,o2 be Object of EnsCat X;
  assume
A1: <^o1,o2^> <> {};
  let A be Morphism of o1,o2, F be Function of o1,o2;
  assume
A2: F = A;
  per cases;
  suppose
    o2 <> {};
    then
A3: dom F = o1 by FUNCT_2:def 1;
    thus A is mono implies F is one-to-one
    proof
      set o = o1;
      assume
A4:   A is mono;
      assume not F is one-to-one;
      then consider x1,x2 be object such that
A5:   x1 in dom F and
A6:   x2 in dom F and
A7:   F.x1 = F.x2 and
A8:   x1 <> x2 by FUNCT_1:def 4;
      set C = o --> x2;
      set B = o --> x1;
A9:   dom C = o by FUNCOP_1:13;
A10:  rng C c= o1
      proof
        let y be object;
        assume y in rng C;
        then ex x be object st x in dom C & C.x = y by FUNCT_1:def 3;
        hence thesis by A3,A6,A9,FUNCOP_1:7;
      end;
      then
A11:  dom (F * C) = o by A3,A9,RELAT_1:27;
      C in Funcs(o,o1) by A9,A10,FUNCT_2:def 2;
      then reconsider C1=C as Morphism of o,o1 by ALTCAT_1:def 14;
      set o9 = the Element of o;
A12:  <^o,o1^> <> {} by ALTCAT_1:19;
      B.o9 = x1 by A3,A5,FUNCOP_1:7;
      then
A13:  B.o9 <> C.o9 by A3,A5,A8,FUNCOP_1:7;
A14:  dom B = o by FUNCOP_1:13;
A15:  rng B c= o1
      proof
        let y be object;
        assume y in rng B;
        then ex x be object st x in dom B & B.x = y by FUNCT_1:def 3;
        hence thesis by A3,A5,A14,FUNCOP_1:7;
      end;
      then B in Funcs(o,o1) by A14,FUNCT_2:def 2;
      then reconsider B1=B as Morphism of o,o1 by ALTCAT_1:def 14;
A16:  dom (F * B) = o by A3,A14,A15,RELAT_1:27;
      now
        let z be object;
        assume
A17:    z in o;
        hence (F * B).z = F.(B.z) by A16,FUNCT_1:12
          .= F.x2 by A7,A17,FUNCOP_1:7
          .= F.(C.z) by A17,FUNCOP_1:7
          .= (F * C).z by A11,A17,FUNCT_1:12;
      end;
      then F * B = F * C by A16,A11,FUNCT_1:2;
      then A * B1 = F * C by A1,A2,A12,ALTCAT_1:16
        .= A * C1 by A1,A2,A12,ALTCAT_1:16;
      hence contradiction by A4,A12,A13;
    end;
    thus F is one-to-one implies A is mono
    proof
      assume
A18:  F is one-to-one;
      let o be Object of EnsCat X;
      assume
A19:  <^o,o1^> <> {};
      then
A20:  <^o,o2^> <> {} by A1,ALTCAT_1:def 2;
      let B,C be Morphism of o,o1;
A21:  <^o,o1^> = Funcs(o,o1) by ALTCAT_1:def 14;
      then consider B1 be Function such that
A22:  B1 = B and
A23:  dom B1 = o and
A24:  rng B1 c= o1 by A19,FUNCT_2:def 2;
      consider C1 be Function such that
A25:  C1 = C and
A26:  dom C1 = o and
A27:  rng C1 c= o1 by A19,A21,FUNCT_2:def 2;
      assume A * B = A * C;
      then
A28:  F * B1 = A * C by A1,A2,A19,A22,A20,ALTCAT_1:16
        .= F * C1 by A1,A2,A19,A25,A20,ALTCAT_1:16;
      now
        let z be object;
        assume
A29:    z in o;
        then F.(B1.z) = (F*B1).z by A23,FUNCT_1:13;
        then
A30:    F.(B1.z) = F.(C1.z) by A26,A28,A29,FUNCT_1:13;
        B1.z in rng B1 & C1.z in rng C1 by A23,A26,A29,FUNCT_1:def 3;
        hence B1.z = C1.z by A3,A18,A24,A27,A30,FUNCT_1:def 4;
      end;
      hence thesis by A22,A23,A25,A26,FUNCT_1:2;
    end;
  end;
  suppose
A31: o2 = {};
     then F = {};
    hence A is mono implies F is one-to-one;
    thus F is one-to-one implies A is mono
    proof
      set x = the Element of Funcs(o1,o2);
      assume F is one-to-one;
      let o be Object of EnsCat X;
      assume
A32:  <^o,o1^> <> {};
      <^o1,o2^> = Funcs(o1,o2) by ALTCAT_1:def 14;
      then consider f be Function such that
      f = x and
A33:  dom f = o1 and
A34:  rng f c= o2 by A1,FUNCT_2:def 2;
      let B,C be Morphism of o,o1;
A35:  <^o,o1^> = Funcs(o,o1) by ALTCAT_1:def 14;
      then consider B1 be Function such that
A36:  B1 = B and
A37:  dom B1 = o and
A38:  rng B1 c= o1 by A32,FUNCT_2:def 2;
      rng f = {} by A31,A34,XBOOLE_1:3;
      then dom f = {} by RELAT_1:42;
      then
A39:  rng B1 = {} by A38,A33,XBOOLE_1:3;
      then
A40:  dom B1 = {} by RELAT_1:42;
      assume A * B = A * C;
      consider C1 be Function such that
A41:  C1 = C and
A42:  dom C1 = o and
      rng C1 c= o1 by A32,A35,FUNCT_2:def 2;
      B1 = {} by A39,RELAT_1:41
        .= C1 by A37,A42,A40,RELAT_1:41;
      hence thesis by A36,A41;
    end;
  end;
end;

theorem
  for X being non empty with_non-empty_elements set for o1,o2 being
  Object of EnsCat X st <^o1,o2^> <> {} for A being Morphism of o1,o2, F being
  Function of o1,o2 st F = A holds A is epi iff F is onto
proof
  let X be non empty with_non-empty_elements set, o1,o2 be Object of EnsCat X;
  assume
A1: <^o1,o2^> <> {};
  let A be Morphism of o1,o2, F be Function of o1,o2;
  assume
A2: F = A;
  per cases;
  suppose
A3: for x be set st x in X holds x is trivial;
    thus A is epi implies F is onto
    proof
      assume A is epi;
      now
        per cases;
        suppose
A4:       o2 = {};
           then F = {};
          hence thesis by A4,FUNCT_2:def 3,RELAT_1:38;
        end;
        suppose
A5:       o2 <> {};
A6:       o1 is Element of X by ALTCAT_1:def 14;
          then o1 is trivial by A3;
          then consider z be object such that
A7:       o1 = {z} by A6,ZFMISC_1:131;
          dom F = {z} by A5,A7,FUNCT_2:def 1;
          then
A8:       rng F <> {} by RELAT_1:42;
          o2 is Element of X by ALTCAT_1:def 14;
          then o2 is trivial by A3;
          then consider y be object such that
A9:       o2 = {y} by A5,ZFMISC_1:131;
          rng F c= {y} by A9,RELAT_1:def 19;
          then rng F = {y} by A8,ZFMISC_1:33;
          hence thesis by A9,FUNCT_2:def 3;
        end;
      end;
      hence thesis;
    end;
    thus F is onto implies A is epi
    proof
      assume
A10:  F is onto;
      let o be Object of EnsCat X;
      assume
A11:  <^o2,o^> <> {};
      then
A12:  <^o1,o^> <> {} by A1,ALTCAT_1:def 2;
      let B,C be Morphism of o2,o;
A13:  <^o2,o^> = Funcs(o2,o) by ALTCAT_1:def 14;
      then consider B1 be Function such that
A14:  B1 = B and
A15:  dom B1 = o2 and
      rng B1 c= o by A11,FUNCT_2:def 2;
      consider C1 be Function such that
A16:  C1 = C and
A17:  dom C1 = o2 and
      rng C1 c= o by A11,A13,FUNCT_2:def 2;
      assume B * A = C * A;
      then
A18:  B1 * F = C * A by A1,A2,A11,A14,A12,ALTCAT_1:16
        .= C1 * F by A1,A2,A11,A16,A12,ALTCAT_1:16;
      now
        assume B1 <> C1;
        then consider z be object such that
A19:    z in o2 and
A20:    B1.z <> C1.z by A15,A17,FUNCT_1:2;
        z in rng F by A10,A19,FUNCT_2:def 3;
        then consider x be object such that
A21:    x in dom F and
A22:    F.x = z by FUNCT_1:def 3;
        B1.(F.x) = (B1*F).x by A21,FUNCT_1:13;
        hence contradiction by A18,A20,A21,A22,FUNCT_1:13;
      end;
      hence thesis by A14,A16;
    end;
  end;
  suppose
A23: ex x be set st x in X & x is non trivial;
    now
      per cases;
      suppose
A24:    o2 <> {};
        consider o be set such that
A25:    o in X and
A26:    o is non trivial by A23;
        reconsider o as Object of EnsCat X by A25,ALTCAT_1:def 14;
A27:    dom F = o1 by A24,FUNCT_2:def 1;
        thus A is epi implies F is onto
        proof
          set k = the Element of o;
A28:      rng F c= o2 by RELAT_1:def 19;
          reconsider ok = (o\{k}) as non empty set by A26,ZFMISC_1:139;
          assume that
A29:      A is epi and
A30:      not F is onto;
          rng F <> o2 by A30,FUNCT_2:def 3;
          then not o2 c= rng F by A28,XBOOLE_0:def 10;
          then consider y be object such that
A31:      y in o2 and
A32:      not y in rng F;
          set C = o2 --> k;
A33:      dom C = o2 by FUNCOP_1:13;
A34:      o <> {} by A25;
          then
A35:      k in o;
          rng C c= o
          proof
            let y be object;
            assume y in rng C;
            then ex x be object st x in dom C & C.x = y by FUNCT_1:def 3;
            hence thesis by A35,A33,FUNCOP_1:7;
          end;
          then C in Funcs(o2,o) by A33,FUNCT_2:def 2;
          then reconsider C1=C as Morphism of o2,o by ALTCAT_1:def 14;
          set l = the Element of ok;
A36:      not l in {k} by XBOOLE_0:def 5;
          reconsider l as Element of o by XBOOLE_0:def 5;
A37:      k <> l by A36,TARSKI:def 1;
          deffunc G(object) = IFEQ($1,y,l,k);
          consider B be Function such that
A38:      dom B = o2 and
A39:      for x be object st x in o2 holds B.x = G(x) from FUNCT_1:sch 3;
A40:      dom (B*F) = o1 by A27,A28,A38,RELAT_1:27;
A41:      rng B c= o
          proof
            let y1 be object;
            assume y1 in rng B;
            then consider x be object such that
A42:        x in dom B & B.x = y1 by FUNCT_1:def 3;
            per cases;
            suppose
A43:          x = y;
              y1 = IFEQ(x,y,l,k) by A38,A39,A42
                .= l by A43,FUNCOP_1:def 8;
             hence thesis by A34;
            end;
            suppose
A44:          x <> y;
              y1 = IFEQ(x,y,l,k) by A38,A39,A42
                .= k by A44,FUNCOP_1:def 8;
              hence thesis by A34;
            end;
          end;
          then
A45:      B in Funcs(o2,o) by A38,FUNCT_2:def 2;
          then
A46:      B in <^o2,o^> by ALTCAT_1:def 14;
          reconsider B1=B as Morphism of o2,o by A45,ALTCAT_1:def 14;
          for z be object holds z in rng(B*F) implies z in rng B
by FUNCT_1:14;
          then rng (B*F) c= rng B;
          then rng (B*F) c= o by A41;
          then (B*F) in Funcs(o1,o) by A40,FUNCT_2:def 2;
          then
A47:      (B*F) in <^o1,o^> by ALTCAT_1:def 14;
          B.y = IFEQ(y,y,l,k) by A31,A39
            .= l by FUNCOP_1:def 8;
          then
A48:      not B = C by A31,A37,FUNCOP_1:7;
A49:      dom (C*F) = o1 by A27,A28,A33,RELAT_1:27;
          now
            let z be object;
            assume
A50:        z in o1;
            then
A51:        F.z in rng F by A27,FUNCT_1:def 3;
            then
A52:        B.(F.z) = IFEQ((F.z),y,l,k) by A28,A39
              .= k by A32,A51,FUNCOP_1:def 8;
            thus (B * F).z = B.(F.z) by A40,A50,FUNCT_1:12
              .= C.(F.z) by A28,A51,A52,FUNCOP_1:7
              .= (C * F).z by A49,A50,FUNCT_1:12;
          end;
          then B * F = C * F by A40,A49,FUNCT_1:2;
          then B1 * A = C * F by A1,A2,A46,A47,ALTCAT_1:16
            .= C1 * A by A1,A2,A46,A47,ALTCAT_1:16;
          hence contradiction by A29,A48,A46;
        end;
        thus F is onto implies A is epi
        proof
          assume
A53:      F is onto;
          let o be Object of EnsCat X;
          assume
A54:      <^o2,o^> <> {};
          then
A55:      <^o1,o^> <> {} by A1,ALTCAT_1:def 2;
          let B,C be Morphism of o2,o;
A56:      <^o2,o^> = Funcs(o2,o) by ALTCAT_1:def 14;
          then consider B1 be Function such that
A57:      B1 = B and
A58:      dom B1 = o2 and
          rng B1 c= o by A54,FUNCT_2:def 2;
          consider C1 be Function such that
A59:      C1 = C and
A60:      dom C1 = o2 and
          rng C1 c= o by A54,A56,FUNCT_2:def 2;
          assume B * A = C * A;
          then
A61:      B1 * F = C * A by A1,A2,A54,A57,A55,ALTCAT_1:16
            .= C1 * F by A1,A2,A54,A59,A55,ALTCAT_1:16;
          now
            assume B1 <> C1;
            then consider z be object such that
A62:        z in o2 and
A63:        B1.z <> C1.z by A58,A60,FUNCT_1:2;
            z in rng F by A53,A62,FUNCT_2:def 3;
            then consider x be object such that
A64:        x in dom F and
A65:        F.x = z by FUNCT_1:def 3;
            B1.(F.x) = (B1*F).x by A64,FUNCT_1:13;
            hence contradiction by A61,A63,A64,A65,FUNCT_1:13;
          end;
          hence thesis by A57,A59;
        end;
      end;
      suppose
A66:    o2 = {};
         then F = {};
        hence A is epi implies F is onto by A66,FUNCT_2:def 3,RELAT_1:38;
        thus F is onto implies A is epi
        proof
          assume F is onto;
          let o be Object of EnsCat X;
          assume
A67:      <^o2,o^> <> {};
          let B,C be Morphism of o2,o;
A68:      <^o2,o^> = Funcs(o2,o) by ALTCAT_1:def 14;
          then consider B1 be Function such that
A69:      B1 = B and
A70:      dom B1 = o2 and
          rng B1 c= o by A67,FUNCT_2:def 2;
A71:      ex C1 be Function st C1 = C & dom C1 = o2 & rng C1 c= o by A67,A68,
FUNCT_2:def 2;
          assume B * A = C * A;
          B1 = {} by A66,A70,RELAT_1:41;
          hence thesis by A66,A69,A71,RELAT_1:41;
        end;
      end;
    end;
    hence thesis;
  end;
end;

theorem Th15:
  for C being category, o1,o2 being Object of C st <^o1,o2^> <> {}
  & <^o2,o1^> <> {} for A being Morphism of o1,o2 st A is retraction holds A is
  epi
proof
  let C be category, o1,o2 be Object of C;
  assume
A1: <^o1,o2^> <> {} & <^o2,o1^> <> {};
  let A be Morphism of o1,o2;
  assume A is retraction;
  then consider R being Morphism of o2,o1 such that
A2: R is_right_inverse_of A;
  let o be Object of C;
  assume
A3: <^o2,o^> <> {};
  let B,C be Morphism of o2,o;
  assume
A4: B * A = C * A;
  thus B = B * idm o2 by A3,ALTCAT_1:def 17
    .= B * (A * R) by A2
    .= C * A * R by A1,A3,A4,ALTCAT_1:21
    .= C * (A * R) by A1,A3,ALTCAT_1:21
    .= C * idm o2 by A2
    .= C by A3,ALTCAT_1:def 17;
end;

theorem Th16:
  for C being category, o1,o2 being Object of C st <^o1,o2^> <> {}
& <^o2,o1^> <> {} for A being Morphism of o1,o2 st A is coretraction holds A is
  mono
proof
  let C be category, o1,o2 be Object of C;
  assume
A1: <^o1,o2^> <> {} & <^o2,o1^> <> {};
  let A be Morphism of o1,o2;
  assume A is coretraction;
  then consider R being Morphism of o2,o1 such that
A2: R is_left_inverse_of A;
  let o be Object of C;
  assume
A3: <^o,o1^> <> {};
  let B,C be Morphism of o,o1;
  assume
A4: A * B = A * C;
  thus B = idm o1 * B by A3,ALTCAT_1:20
    .= R * A * B by A2
    .= R * (A * C) by A1,A3,A4,ALTCAT_1:21
    .= R * A * C by A1,A3,ALTCAT_1:21
    .= idm o1 * C by A2
    .= C by A3,ALTCAT_1:20;
end;

theorem
  for C being category, o1,o2 being Object of C st <^o1,o2^> <> {} & <^
  o2,o1^> <> {} for A being Morphism of o1,o2 st A is iso holds A is mono epi
proof
  let C be category;
  let o1, o2 be Object of C such that
A1: <^o1,o2^> <> {} & <^o2,o1^> <> {};
  let A be Morphism of o1, o2;
  assume A is iso;
  then
A2: A is retraction & A is coretraction by A1,Th6;
A3: for o being Object of C st <^o2,o^> <> {} for B, C being Morphism of o2
  , o st B * A = C * A holds B = C
  proof
    let o be Object of C such that
A4: <^o2,o^> <> {};
    let B, C be Morphism of o2, o;
    assume B * A = C * A;
    then B * (A * A") = (C * A) * A" by A1,A4,ALTCAT_1:21;
    then B * idm o2 = (C * A) * A" by A1,A2,Th2;
    then B * idm o2 = C * (A * A") by A1,A4,ALTCAT_1:21;
    then B * idm o2 = C * idm o2 by A1,A2,Th2;
    then B = C * idm o2 by A4,ALTCAT_1:def 17;
    hence thesis by A4,ALTCAT_1:def 17;
  end;
  for o being Object of C st <^o,o1^> <> {} for B, C being Morphism of o,
  o1 st A * B = A * C holds B = C
  proof
    let o be Object of C such that
A5: <^o,o1^> <> {};
    let B, C be Morphism of o, o1;
    assume A * B = A * C;
    then (A" * A) * B = A" * (A * C) by A1,A5,ALTCAT_1:21;
    then idm o1 * B = A" * (A * C) by A1,A2,Th2;
    then idm o1 * B = (A" * A) * C by A1,A5,ALTCAT_1:21;
    then idm o1 * B = idm o1 * C by A1,A2,Th2;
    then B = idm o1 * C by A5,ALTCAT_1:20;
    hence thesis by A5,ALTCAT_1:20;
  end;
  hence thesis by A3;
end;

theorem Th18:
  for C being category, o1,o2,o3 being Object of C st <^o1,o2^> <>
  {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} for A being Morphism of o1,o2, B being
Morphism of o2,o3 st A is retraction & B is retraction holds B*A is retraction
proof
  let C be category, o1,o2,o3 be Object of C;
  assume that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {} and
A3: <^o3,o1^> <> {};
A4: <^o2,o1^> <> {} by A2,A3,ALTCAT_1:def 2;
A5: <^o3,o2^> <> {} by A1,A3,ALTCAT_1:def 2;
  let A be Morphism of o1,o2, B be Morphism of o2,o3;
  assume that
A6: A is retraction and
A7: B is retraction;
  consider A1 being Morphism of o2,o1 such that
A8: A1 is_right_inverse_of A by A6;
  consider B1 being Morphism of o3,o2 such that
A9: B1 is_right_inverse_of B by A7;
  consider G being Morphism of o3,o1 such that
A10: G = A1 * B1;
  take G;
  (B * A) * G = B * (A * (A1 * B1)) by A1,A2,A3,A10,ALTCAT_1:21
    .= B * ((A * A1) * B1) by A1,A4,A5,ALTCAT_1:21
    .= B * (idm o2 * B1) by A8
    .= B * B1 by A5,ALTCAT_1:20
    .= idm o3 by A9;
  hence thesis;
end;

theorem Th19:
  for C being category, o1,o2,o3 being Object of C st <^o1,o2^> <>
  {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} for A being Morphism of o1,o2, B being
  Morphism of o2,o3 st A is coretraction & B is coretraction holds B*A is
  coretraction
proof
  let C be category, o1,o2,o3 be Object of C;
  assume that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {} and
A3: <^o3,o1^> <> {};
A4: <^o2,o1^> <> {} by A2,A3,ALTCAT_1:def 2;
A5: <^o3,o2^> <> {} by A1,A3,ALTCAT_1:def 2;
  let A be Morphism of o1,o2, B be Morphism of o2,o3;
  assume that
A6: A is coretraction and
A7: B is coretraction;
  consider A1 being Morphism of o2,o1 such that
A8: A1 is_left_inverse_of A by A6;
  consider B1 being Morphism of o3,o2 such that
A9: B1 is_left_inverse_of B by A7;
  consider G being Morphism of o3,o1 such that
A10: G = A1 * B1;
  take G;
A11: <^o2,o2^> <> {} by ALTCAT_1:19;
  G * (B * A) = ((A1 * B1) * B) * A by A1,A2,A3,A10,ALTCAT_1:21
    .= (A1 * (B1 * B)) * A by A2,A4,A5,ALTCAT_1:21
    .= (A1 * idm o2) * A by A9
    .= A1 * (idm o2 *A) by A1,A4,A11,ALTCAT_1:21
    .= A1 * A by A1,ALTCAT_1:20
    .= idm o1 by A8;
  hence thesis;
end;

theorem Th20:
  for C being category, o1, o2 being Object of C, A being Morphism
  of o1,o2 st A is retraction & A is mono & <^o1,o2^> <> {} & <^o2,o1^> <> {}
  holds A is iso
proof
  let C be category, o1, o2 be Object of C, A be Morphism of o1,o2;
  assume that
A1: A is retraction and
A2: A is mono and
A3: <^o1,o2^> <> {} and
A4: <^o2,o1^> <> {};
  consider B being Morphism of o2,o1 such that
A5: B is_right_inverse_of A by A1;
  A * B * A = (idm o2) * A by A5;
  then A * (B * A) = (idm o2) * A by A3,A4,ALTCAT_1:21;
  then A * (B * A) = A by A3,ALTCAT_1:20;
  then
A6: <^o1,o1^> <> {} & A * (B * A) = A * idm o1 by A3,ALTCAT_1:19,def 17;
  then B * A = idm o1 by A2;
  then
A7: B is_left_inverse_of A;
  then
A8: A is coretraction;
  then
A9: A*A" = A * B by A1,A3,A4,A5,A7,Def4
    .= idm o2 by A5;
  A"*A = B * A by A1,A3,A4,A5,A7,A8,Def4
    .= idm o1 by A2,A6;
  hence thesis by A9;
end;

theorem
  for C being category, o1, o2 being Object of C, A being Morphism of o1
, o2 st A is coretraction & A is epi & <^o1,o2^> <> {} & <^o2,o1^> <> {} holds
  A is iso
proof
  let C be category, o1, o2 be Object of C, A be Morphism of o1,o2;
  assume that
A1: A is coretraction and
A2: A is epi and
A3: <^o1,o2^> <> {} and
A4: <^o2,o1^> <> {};
  consider B being Morphism of o2,o1 such that
A5: B is_left_inverse_of A by A1;
  A * (B * A) = A * (idm o1) by A5;
  then A * (B * A) = A by A3,ALTCAT_1:def 17;
  then A * (B * A) = idm o2 * A by A3,ALTCAT_1:20;
  then
A6: <^o2,o2^> <> {} & (A * B) * A = idm o2 * A by A3,A4,ALTCAT_1:19,21;
  then A * B = idm o2 by A2;
  then
A7: B is_right_inverse_of A;
  then
A8: A is retraction;
  then
A9: A"*A = B * A by A1,A3,A4,A5,A7,Def4
    .= idm o1 by A5;
  A*A" = A * B by A1,A3,A4,A5,A7,A8,Def4
    .= idm o2 by A2,A6;
  hence thesis by A9;
end;

theorem
  for C being category, o1,o2,o3 being Object of C, A being Morphism of
o1, o2, B being Morphism of o2,o3 st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,
  o1^> <> {} & B * A is retraction holds B is retraction
proof
  let C be category, o1,o2,o3 be Object of C, A be Morphism of o1,o2, B be
  Morphism of o2,o3;
  assume
A1: <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {};
  assume B * A is retraction;
  then consider G be Morphism of o3,o1 such that
A2: G is_right_inverse_of (B*A);
  (B * A) * G = idm o3 by A2;
  then B * (A * G) = idm o3 by A1,ALTCAT_1:21;
  then A * G is_right_inverse_of B;
  hence thesis;
end;

theorem
  for C being category, o1,o2,o3 being Object of C, A being Morphism of
o1, o2, B being Morphism of o2,o3 st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,
  o1^> <> {} & B * A is coretraction holds A is coretraction
proof
  let C be category, o1,o2,o3 be Object of C, A be Morphism of o1,o2, B be
  Morphism of o2,o3;
  assume
A1: <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {};
  assume B * A is coretraction;
  then consider G be Morphism of o3,o1 such that
A2: G is_left_inverse_of (B * A);
A3: (G * B) * A = G * (B * A) by A1,ALTCAT_1:21;
  G * (B * A) = idm o1 by A2;
  then G * B is_left_inverse_of A by A3;
  hence thesis;
end;

theorem
  for C being category st for o1,o2 being Object of C, A1 being Morphism
  of o1,o2 holds A1 is retraction holds for a,b being Object of C,A being
  Morphism of a,b st <^a,b^> <> {} & <^b,a^> <> {} holds A is iso
proof
  let C be category;
  assume
A1: for o1,o2 being Object of C, A1 being Morphism of o1,o2 holds A1 is
  retraction;
  thus for a,b being Object of C, A being Morphism of a,b st <^a,b^> <> {} &
  <^b,a^> <> {} holds A is iso
  proof
    let a,b be Object of C;
    let A be Morphism of a,b;
    assume that
A2: <^a,b^> <> {} and
A3: <^b,a^> <> {};
A4: A is retraction by A1;
    A is coretraction
    proof
      consider A1 be Morphism of b,a such that
A5:   A1 is_right_inverse_of A by A4;
      A1 * (A * A1) =A1 * idm b by A5;
      then A1 * (A * A1) =A1 by A3,ALTCAT_1:def 17;
      then (A1 * A) * A1 =A1 by A2,A3,ALTCAT_1:21;
      then
A6:   (A1 * A) * A1 =idm a * A1 by A3,ALTCAT_1:20;
      A1 is epi & <^a,a^> <> {} by A1,A2,A3,Th15,ALTCAT_1:19;
      then (A1 * A) =idm a by A6;
      then A1 is_left_inverse_of A;
      hence thesis;
    end;
    hence thesis by A2,A3,A4,Th6;
  end;
end;

registration
  let C be with_units non empty AltCatStr, o be Object of C;
  cluster mono epi retraction coretraction for Morphism of o,o;
  existence
  proof
    take idm o;
    thus thesis by Lm1,Th1;
  end;
end;

registration
  let C be category, o be Object of C;
  cluster mono epi iso retraction coretraction for Morphism of o,o;
  existence
  proof
    take I = idm o;
    <^o,o^> <> {} & I is retraction coretraction by Th1,ALTCAT_1:19;
    hence thesis by Th15,Th16,Th20;
  end;
end;

registration
  let C be category, o be Object of C, A, B be mono Morphism of o,o;
  cluster A * B -> mono;
  coherence
  proof
    <^o,o^> <> {} by ALTCAT_1:19;
    hence thesis by Th9;
  end;
end;

registration
  let C be category, o be Object of C, A, B be epi Morphism of o,o;
  cluster A * B -> epi;
  coherence
  proof
    <^o,o^> <> {} by ALTCAT_1:19;
    hence thesis by Th10;
  end;
end;

registration
  let C be category, o be Object of C, A, B be iso Morphism of o,o;
  cluster A * B -> iso;
  coherence
  proof
    <^o,o^> <> {} by ALTCAT_1:19;
    hence thesis by Th7;
  end;
end;

registration
  let C be category, o be Object of C, A, B be retraction Morphism of o,o;
  cluster A * B -> retraction;
  coherence
  proof
    <^o,o^> <> {} by ALTCAT_1:19;
    hence thesis by Th18;
  end;
end;

registration
  let C be category, o be Object of C, A, B be coretraction Morphism of o,o;
  cluster A * B -> coretraction;
  coherence
  proof
    <^o,o^> <> {} by ALTCAT_1:19;
    hence thesis by Th19;
  end;
end;

definition
  let C be AltGraph, o be Object of C;
  attr o is initial means

  for o1 being Object of C holds ex M being
  Morphism of o,o1 st M in <^o,o1^> & <^o,o1^> is trivial;
end;

theorem
  for C being AltGraph, o being Object of C holds o is initial iff for
  o1 being Object of C holds ex M being Morphism of o,o1 st M in <^o,o1^> & for
  M1 being Morphism of o,o1 st M1 in <^o,o1^> holds M = M1
proof
  let C be AltGraph, o be Object of C;
  thus o is initial implies for o1 being Object of C holds ex M being Morphism
  of o,o1 st M in <^o,o1^> & for M1 being Morphism of o,o1 st M1 in <^o,o1^>
  holds M = M1
  proof
    assume
A1: o is initial;
    let o1 be Object of C;
    consider M being Morphism of o,o1 such that
A2: M in <^o,o1^> and
A3: <^o,o1^> is trivial by A1;
    ex i be object st <^o,o1^> = { i } by A2,A3,ZFMISC_1:131;
    then <^o,o1^> = {M} by TARSKI:def 1;
    then for M1 being Morphism of o,o1 st M1 in <^o,o1^> holds M = M1 by
TARSKI:def 1;
    hence thesis by A2;
  end;
  assume
A4: for o1 being Object of C holds ex M being Morphism of o,o1 st M in
  <^o,o1^> & for M1 being Morphism of o,o1 st M1 in <^o,o1^> holds M = M1;
  let o1 be Object of C;
  consider M being Morphism of o,o1 such that
A5: M in <^o,o1^> and
A6: for M1 being Morphism of o,o1 st M1 in <^o,o1^> holds M = M1 by A4;
A7: <^o,o1^> c= {M}
  proof
    let x be object;
    assume
A8: x in <^o,o1^>;
    then reconsider M1 = x as Morphism of o,o1;
    M1 = M by A6,A8;
    hence thesis by TARSKI:def 1;
  end;
  {M} c= <^o,o1^>
  by A5,TARSKI:def 1;
  then <^o,o1^> = {M} by A7,XBOOLE_0:def 10;
  hence thesis;
end;

theorem Th26:
  for C being category, o1,o2 being Object of C st o1 is initial &
  o2 is initial holds o1,o2 are_iso
proof
  let C be category, o1,o2 be Object of C such that
A1: o1 is initial and
A2: o2 is initial;
  ex N being Morphism of o2,o2 st N in <^o2,o2^> & <^o2,o2^> is trivial by A2;
  then consider y being object such that
A3: <^o2,o2^> = {y} by ZFMISC_1:131;
  consider M2 being Morphism of o2,o1 such that
A4: M2 in <^o2,o1^> and
  <^o2,o1^> is trivial by A2;
  consider M1 being Morphism of o1,o2 such that
A5: M1 in <^o1,o2^> and
  <^o1,o2^> is trivial by A1;
  thus <^o1,o2^> <> {} & <^o2,o1^> <> {} by A5,A4;
  M1 * M2 = y & idm o2 = y by A3,TARSKI:def 1;
  then M2 is_right_inverse_of M1;
  then
A6: M1 is retraction;
  ex M being Morphism of o1,o1 st M in <^o1,o1^> & <^o1,o1^> is trivial by A1;
  then consider x being object such that
A7: <^o1,o1^> = {x} by ZFMISC_1:131;
  M2 * M1 = x & idm o1 = x by A7,TARSKI:def 1;
  then M2 is_left_inverse_of M1;
  then M1 is coretraction;
  then M1 is iso by A5,A4,A6,Th6;
  hence thesis;
end;

definition
  let C be AltGraph, o be Object of C;
  attr o is terminal means

  for o1 being Object of C holds ex M being
  Morphism of o1,o st M in <^o1,o^> & <^o1,o^> is trivial;
end;

theorem
  for C being AltGraph, o being Object of C holds o is terminal iff for
  o1 being Object of C holds ex M being Morphism of o1,o st M in <^o1,o^> & for
  M1 being Morphism of o1,o st M1 in <^o1,o^> holds M = M1
proof
  let C be AltGraph, o be Object of C;
  thus o is terminal implies for o1 being Object of C holds ex M being
Morphism of o1,o st M in <^o1,o^> & for M1 being Morphism of o1,o st M1 in <^o1
  ,o^> holds M = M1
  proof
    assume
A1: o is terminal;
    let o1 be Object of C;
    consider M being Morphism of o1,o such that
A2: M in <^o1,o^> and
A3: <^o1,o^> is trivial by A1;
    ex i be object st <^o1,o^> = { i } by A2,A3,ZFMISC_1:131;
    then <^o1,o^> = {M} by TARSKI:def 1;
    then for M1 being Morphism of o1,o st M1 in <^o1,o^> holds M = M1 by
TARSKI:def 1;
    hence thesis by A2;
  end;
  assume
A4: for o1 being Object of C holds ex M being Morphism of o1,o st M in
  <^o1,o^> & for M1 being Morphism of o1,o st M1 in <^o1,o^> holds M = M1;
  let o1 be Object of C;
  consider M being Morphism of o1,o such that
A5: M in <^o1,o^> and
A6: for M1 being Morphism of o1,o st M1 in <^o1,o^> holds M = M1 by A4;
A7: <^o1,o^> c= {M}
  proof
    let x be object;
    assume
A8: x in <^o1,o^>;
    then reconsider M1 = x as Morphism of o1,o;
    M1 = M by A6,A8;
    hence thesis by TARSKI:def 1;
  end;
  {M} c= <^o1,o^>
  by A5,TARSKI:def 1;
  then <^o1,o^> = {M} by A7,XBOOLE_0:def 10;
  hence thesis;
end;

theorem
  for C being category, o1,o2 being Object of C st o1 is terminal & o2
  is terminal holds o1,o2 are_iso
proof
  let C be category, o1,o2 be Object of C;
  assume that
A1: o1 is terminal and
A2: o2 is terminal;
  ex M being Morphism of o1,o1 st M in <^o1,o1^> & <^o1,o1^> is trivial by A1;
  then consider x being object such that
A3: <^o1,o1^> = {x} by ZFMISC_1:131;
  consider M2 being Morphism of o2,o1 such that
A4: M2 in <^o2,o1^> and
  <^o2,o1^> is trivial by A1;
  consider M1 being Morphism of o1,o2 such that
A5: M1 in <^o1,o2^> and
  <^o1,o2^> is trivial by A2;
  thus <^o1,o2^> <> {} & <^o2,o1^> <> {} by A5,A4;
  M2 * M1 = x by A3,TARSKI:def 1;
  then M2 * M1 = idm o1 by A3,TARSKI:def 1;
  then M2 is_left_inverse_of M1;
  then
A6: M1 is coretraction;
  ex N being Morphism of o2,o2 st N in <^o2,o2^> & <^o2,o2^> is trivial by A2;
  then consider y being object such that
A7: <^o2,o2^> = {y} by ZFMISC_1:131;
  M1 * M2 = y by A7,TARSKI:def 1;
  then M1 * M2 = idm o2 by A7,TARSKI:def 1;
  then M2 is_right_inverse_of M1;
  then M1 is retraction;
  then M1 is iso by A5,A4,A6,Th6;
  hence thesis;
end;

definition
  let C be AltGraph, o be Object of C;
  attr o is _zero means

  o is initial terminal;
end;

theorem
  for C being category, o1,o2 being Object of C st o1 is _zero & o2 is
  _zero holds o1,o2 are_iso
by Th26;

definition
  let C be non empty AltCatStr, o1, o2 be Object of C, M be Morphism of o1,o2;
  attr M is _zero means

  for o being Object of C st o is _zero for A
  being Morphism of o1,o, B being Morphism of o,o2 holds M = B*A;
end;

theorem
  for C being category, o1,o2,o3 being Object of C for M1 being Morphism
of o1,o2, M2 being Morphism of o2,o3 st M1 is _zero & M2 is _zero holds M2 * M1
  is _zero
proof
  let C be category, o1,o2,o3 be Object of C, M1 be Morphism of o1,o2, M2 be
  Morphism of o2,o3;
  assume that
A1: M1 is _zero and
A2: M2 is _zero;
  let o be Object of C;
  assume
A3: o is _zero;
  then
A4: o is initial;
  then consider B1 being Morphism of o,o2 such that
A5: B1 in <^o,o2^> and
  <^o,o2^> is trivial;
  let A be Morphism of o1,o, B be Morphism of o,o3;
  consider B2 being Morphism of o,o3 such that
A6: B2 in <^o,o3^> and
A7: <^o,o3^> is trivial by A4;
  consider y being object such that
A8: <^o,o3^> = {y} by A6,A7,ZFMISC_1:131;
A9: o is terminal by A3;
  then consider A1 being Morphism of o1,o such that
A10: A1 in <^o1,o^> and
A11: <^o1,o^> is trivial;
  consider x being object such that
A12: <^o1,o^> = {x} by A10,A11,ZFMISC_1:131;
  ex M being Morphism of o,o st M in <^o,o^> & <^o,o^> is trivial by A9;
  then consider z being object such that
A13: <^o,o^> = {z} by ZFMISC_1:131;
  consider A2 being Morphism of o2,o such that
A14: A2 in <^o2,o^> and
  <^o2,o^> is trivial by A9;
A15: idm o = z & A2 * B1 = z by A13,TARSKI:def 1;
A16: B = y & B2 = y by A8,TARSKI:def 1;
A17: A = x & A1 = x by A12,TARSKI:def 1;
A18: <^o2,o3^> <> {} by A6,A14,ALTCAT_1:def 2;
  M2 = B2 * A2 by A2,A3;
  hence M2 * M1 = (B2*A2) * (B1*A1) by A1,A3
    .= B2*A2 * B1*A1 by A5,A10,A18,ALTCAT_1:21
    .= B*(idm o)*A by A5,A6,A14,A17,A16,A15,ALTCAT_1:21
    .= B*A by A6,ALTCAT_1:def 17;
end;