let C be Category; :: thesis:
let c be Object of C; :: thesis:
let p1, p2 be Morphism of C; :: thesis:
set i1 = p1 opp ;
set i2 = p2 opp ;
thus ( c is_a_product_wrt p1,p2 implies c opp is_a_coproduct_wrt p1 opp ,p2 opp ) :: thesis:

proof

assume that
A1: ( dom p1 = c & dom p2 = c ) and
A2: for d being Object of C
for f, g being Morphism of C st f in Hom (d,(cod p1)) & g in Hom (d,(cod p2)) holds
ex h being Morphism of C st
( h in Hom (d,c) & ( for k being Morphism of C st k in Hom (d,c) holds
( ( p1 (*) k = f & p2 (*) k = g ) iff h = k ) ) ) ; :: according to CAT_3:def 15 :: thesis:
reconsider gg = p1 as Morphism of dom p1, cod p1 by CAT_1:4;
A3: Hom ((dom gg),(cod gg)) <> {} by CAT_1:2;
then A4: gg opp = p1 opp by OPPCAT_1:def 6;
thus A5: cod (p1 opp) = c opp by A1, A3, A4, OPPCAT_1:10; :: according to CAT_3:def 18 :: thesis:
reconsider gg = p2 as Morphism of dom p2, cod p2 by CAT_1:4;
A6: Hom ((dom gg),(cod gg)) <> {} by CAT_1:2;
then A7: gg opp = p2 opp by OPPCAT_1:def 6;
thus A8: cod (p2 opp) = c opp by A1, A6, A7, OPPCAT_1:10; :: thesis:
let d be Object of (C opp); :: thesis:
let f, g be Morphism of (C opp); :: thesis:
assume that
A9: f in Hom ((dom (p1 opp)),d) and
A10: g in Hom ((dom (p2 opp)),d) ; :: thesis:
reconsider gg = p2 opp as Morphism of dom (p2 opp), cod (p2 opp) by CAT_1:4;
A11: Hom ((dom gg),(cod gg)) <> {} by CAT_1:2;
opp g in Hom ((opp d),(opp (dom (p2 opp)))) by A10, OPPCAT_1:6;
then A12: opp g in Hom ((opp d),(cod (opp (p2 opp)))) by A11, OPPCAT_1:13;
reconsider gg = p1 opp as Morphism of dom (p1 opp), cod (p1 opp) by CAT_1:4;
A13: Hom ((dom gg),(cod gg)) <> {} by CAT_1:2;
opp f in Hom ((opp d),(opp (dom (p1 opp)))) by A9, OPPCAT_1:6;
then opp f in Hom ((opp d),(cod (opp (p1 opp)))) by A13, OPPCAT_1:13;
then consider h being Morphism of C such that
A14: h in Hom ((opp d),c) and
A15: for k being Morphism of C st k in Hom ((opp d),c) holds
( ( p1 (*) k = opp f & p2 (*) k = opp g ) iff h = k ) by A2, A12;
take h opp ; :: thesis:
h opp in Hom ((c opp),((opp d) opp)) by A14, OPPCAT_1:5;
hence h opp in Hom ((c opp),d) ; :: thesis:
let k be Morphism of (C opp); :: thesis:
assume A16: k in Hom ((c opp),d) ; :: thesis:
then opp k in Hom ((opp d),(opp (c opp))) by OPPCAT_1:6;
then A17: ( ( (opp (p1 opp)) (*) (opp k) = f & (opp (p2 opp)) (*) (opp k) = g ) iff h opp = k ) by A15;
dom k = c opp by A16, CAT_1:1;
then ( ( opp (k (*) (p1 opp)) = f & opp (k (*) (p2 opp)) = g ) iff h opp = k ) by A8, A5, A17, OPPCAT_1:18;
hence ( ( k (*) (p1 opp) = f & k (*) (p2 opp) = g ) iff h opp = k ) ; :: thesis:

end;

assume that
A18: ( cod (p1 opp) = c opp & cod (p2 opp) = c opp ) and
A19: for d being Object of (C opp)
for f, g being Morphism of (C opp) st f in Hom ((dom (p1 opp)),d) & g in Hom ((dom (p2 opp)),d) holds
ex h being Morphism of (C opp) st
( h in Hom ((c opp),d) & ( for k being Morphism of (C opp) st k in Hom ((c opp),d) holds
( ( k (*) (p1 opp) = f & k (*) (p2 opp) = g ) iff h = k ) ) ) ; :: according to CAT_3:def 18 :: thesis:
reconsider gg = p1 as Morphism of dom p1, cod p1 by CAT_1:4;
A20: Hom ((dom gg),(cod gg)) <> {} by CAT_1:2;
then A21: gg opp = p1 opp by OPPCAT_1:def 6;
A22: dom p1 = c opp by A18, A20, A21, OPPCAT_1:10;
reconsider gg = p2 as Morphism of dom p2, cod p2 by CAT_1:4;
A23: Hom ((dom gg),(cod gg)) <> {} by CAT_1:2;
then A24: gg opp = p2 opp by OPPCAT_1:def 6;
A25: dom p2 = c opp by A18, A23, A24, OPPCAT_1:10;
hence ( dom p1 = c & dom p2 = c ) by A22; :: according to CAT_3:def 15 :: thesis:
let d be Object of C; :: thesis:
let f, g be Morphism of C; :: thesis:
assume that
A26: f in Hom (d,(cod p1)) and
A27: g in Hom (d,(cod p2)) ; :: thesis:
g opp in Hom (((cod p2) opp),(d opp)) by A27, OPPCAT_1:5;
then A28: g opp in Hom ((dom (p2 opp)),(d opp)) by A23, A24, OPPCAT_1:12;
f opp in Hom (((cod p1) opp),(d opp)) by A26, OPPCAT_1:5;
then f opp in Hom ((dom (p1 opp)),(d opp)) by A20, A21, OPPCAT_1:12;
then consider h being Morphism of (C opp) such that
A29: h in Hom ((c opp),(d opp)) and
A30: for k being Morphism of (C opp) st k in Hom ((c opp),(d opp)) holds
( ( k (*) (p1 opp) = f opp & k (*) (p2 opp) = g opp ) iff h = k ) by A19, A28;
take opp h ; :: thesis:
thus opp h in Hom (d,c) by A29, OPPCAT_1:5; :: thesis:
let k be Morphism of C; :: thesis:
assume A31: k in Hom (d,c) ; :: thesis:
then k opp in Hom ((c opp),(d opp)) by OPPCAT_1:5;
then A32: ( ( (k opp) (*) (p1 opp) = f opp & (k opp) (*) (p2 opp) = g opp ) iff h = k opp ) by A30;
A33: cod k = c by A31, CAT_1:1;
reconsider ff = p1 as Morphism of dom p1, cod p1 by CAT_1:4;
reconsider gg = k as Morphism of dom k, dom p1 by A33, A22, CAT_1:4;
A34: ( Hom ((dom k),(cod k)) <> {} & Hom ((dom p1),(cod p1)) <> {} ) by CAT_1:2;
then A35: ff opp = p1 opp by OPPCAT_1:def 6;
A36: gg opp = k opp by A34, A33, A22, OPPCAT_1:def 6;
A37: (p1 (*) k) opp = (k opp) (*) (p1 opp) by A22, A34, A33, A35, A36, OPPCAT_1:16;
reconsider ff = p2 as Morphism of dom p2, cod p2 by CAT_1:4;
reconsider gg = k as Morphism of dom k, dom p2 by A33, A25, CAT_1:4;
A38: ( Hom ((dom k),(cod k)) <> {} & Hom ((dom p2),(cod p2)) <> {} ) by CAT_1:2;
then A39: ff opp = p2 opp by OPPCAT_1:def 6;
A40: gg opp = k opp by A38, A33, A25, OPPCAT_1:def 6;
(p2 (*) k) opp = (k opp) (*) (p2 opp) by A38, A33, A39, A40, A25, OPPCAT_1:16;
hence ( ( p1 (*) k = f & p2 (*) k = g ) iff opp h = k ) by A37, A32; :: thesis: