:: 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;