let I be set ; :: thesis: for C being Category

for f being Morphism of C

for F being Projections_family of cod f,I holds (f opp) * (F opp) = (F * f) opp

let C be Category; :: thesis: for f being Morphism of C

for F being Projections_family of cod f,I holds (f opp) * (F opp) = (F * f) opp

let f be Morphism of C; :: thesis: for F being Projections_family of cod f,I holds (f opp) * (F opp) = (F * f) opp

let F be Projections_family of cod f,I; :: thesis: (f opp) * (F opp) = (F * f) opp

for f being Morphism of C

for F being Projections_family of cod f,I holds (f opp) * (F opp) = (F * f) opp

let C be Category; :: thesis: for f being Morphism of C

for F being Projections_family of cod f,I holds (f opp) * (F opp) = (F * f) opp

let f be Morphism of C; :: thesis: for F being Projections_family of cod f,I holds (f opp) * (F opp) = (F * f) opp

let F be Projections_family of cod f,I; :: thesis: (f opp) * (F opp) = (F * f) opp

now :: thesis: for x being set st x in I holds

((f opp) * (F opp)) /. x = ((F * f) opp) /. x

hence
(f opp) * (F opp) = (F * f) opp
by Th1; :: thesis: verum((f opp) * (F opp)) /. x = ((F * f) opp) /. x

let x be set ; :: thesis: ( x in I implies ((f opp) * (F opp)) /. x = ((F * f) opp) /. x )

assume A1: x in I ; :: thesis: ((f opp) * (F opp)) /. x = ((F * f) opp) /. x

then A2: dom (F /. x) = (doms F) /. x by Def1

.= (I --> (cod f)) /. x by Def13

.= cod f by A1, Th2 ;

reconsider ff = f as Morphism of dom f, cod f by CAT_1:4;

reconsider gg = F /. x as Morphism of cod f, cod (F /. x) by A2, CAT_1:4;

A3: ( Hom ((dom f),(cod f)) <> {} & Hom ((dom (F /. x)),(cod (F /. x))) <> {} ) by CAT_1:2;

then A4: ff opp = f opp by OPPCAT_1:def 6;

A5: gg opp = (F /. x) opp by A3, A2, OPPCAT_1:def 6;

thus ((f opp) * (F opp)) /. x = (f opp) (*) ((F opp) /. x) by A1, Def6

.= (f opp) (*) ((F /. x) opp) by A1, Def3

.= (gg (*) ff) opp by A2, A4, A5, A3, OPPCAT_1:16

.= ((F * f) /. x) opp by A1, Def5

.= ((F * f) opp) /. x by A1, Def3 ; :: thesis: verum

end;assume A1: x in I ; :: thesis: ((f opp) * (F opp)) /. x = ((F * f) opp) /. x

then A2: dom (F /. x) = (doms F) /. x by Def1

.= (I --> (cod f)) /. x by Def13

.= cod f by A1, Th2 ;

reconsider ff = f as Morphism of dom f, cod f by CAT_1:4;

reconsider gg = F /. x as Morphism of cod f, cod (F /. x) by A2, CAT_1:4;

A3: ( Hom ((dom f),(cod f)) <> {} & Hom ((dom (F /. x)),(cod (F /. x))) <> {} ) by CAT_1:2;

then A4: ff opp = f opp by OPPCAT_1:def 6;

A5: gg opp = (F /. x) opp by A3, A2, OPPCAT_1:def 6;

thus ((f opp) * (F opp)) /. x = (f opp) (*) ((F opp) /. x) by A1, Def6

.= (f opp) (*) ((F /. x) opp) by A1, Def3

.= (gg (*) ff) opp by A2, A4, A5, A3, OPPCAT_1:16

.= ((F * f) /. x) opp by A1, Def5

.= ((F * f) opp) /. x by A1, Def3 ; :: thesis: verum