deffunc H₁( set ) -> Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:] = the Comp of C | [:((the_hom_sets_of C) . ((($1 `1) `2),($1 `2))),((the_hom_sets_of C) . ((($1 `1) `1),(($1 `1) `2))):];
set Ob3 = [: the carrier of C, the carrier of C, the carrier of C:];
set G = the_hom_sets_of C;
consider o being Function such that
A1: dom o = [: the carrier of C, the carrier of C, the carrier of C:] and
A2: for e being set st e in [: the carrier of C, the carrier of C, the carrier of C:] holds
o . e = H₁(e) from FUNCT_1:sch 3();
reconsider o = o as ManySortedSet of [: the carrier of C, the carrier of C, the carrier of C:] by A1, PARTFUN1:def 2, RELAT_1:def 18;

now

let e be set ; :: thesis: ( e in dom o implies o . e is Function )
assume e in dom o ; :: thesis: o . e is Function
then o . e = the Comp of C | [:((the_hom_sets_of C) . (((e `1) `2),(e `2))),((the_hom_sets_of C) . (((e `1) `1),((e `1) `2))):] by A1, A2;
hence o . e is Function ; :: thesis: verum

end;

then reconsider o = o as ManySortedFunction of [: the carrier of C, the carrier of C, the carrier of C:] by FUNCOP_1:def 6;

now

let e be set ; :: thesis: ( e in [: the carrier of C, the carrier of C, the carrier of C:] implies o . e is Function of ({|(the_hom_sets_of C),(the_hom_sets_of C)|} . e),({|(the_hom_sets_of C)|} . e) )
reconsider f = o . e as Function ;
assume A3: e in [: the carrier of C, the carrier of C, the carrier of C:] ; :: thesis: o . e is Function of ({|(the_hom_sets_of C),(the_hom_sets_of C)|} . e),({|(the_hom_sets_of C)|} . e)
then consider i, j, k being Object of C such that
A4: e = [i,j,k] by DOMAIN_1:3;
reconsider e9 = e as Element of [: the carrier of C, the carrier of C, the carrier of C:] by A3;
A5: ([i,j,k] `1) `2 = e9 `2 by A4, MCART_1:46
.= j by A4, MCART_1:43 ;
A6: [i,j,k] `2 = e9 `3 by A4, MCART_1:46
.= k by A4, MCART_1:43 ;
([i,j,k] `1) `1 = e9 `1 by A4, MCART_1:46
.= i by A4, MCART_1:43 ;
then A7: f = the Comp of C | [:((the_hom_sets_of C) . (j,k)),((the_hom_sets_of C) . (i,j)):] by A2, A4, A5, A6;
A8: ( (the_hom_sets_of C) . (i,j) = Hom (i,j) & (the_hom_sets_of C) . (j,k) = Hom (j,k) ) by Def3;
A9: {|(the_hom_sets_of C)|} . e = {|(the_hom_sets_of C)|} . (i,j,k) by A4, MULTOP_1:def 1
.= (the_hom_sets_of C) . (i,k) by ALTCAT_1:def 3
.= Hom (i,k) by Def3 ;
A10: {|(the_hom_sets_of C),(the_hom_sets_of C)|} . e = {|(the_hom_sets_of C),(the_hom_sets_of C)|} . (i,j,k) by A4, MULTOP_1:def 1
.= [:(Hom (j,k)),(Hom (i,j)):] by A8, ALTCAT_1:def 4 ;

A11: now

assume {|(the_hom_sets_of C)|} . e = {} ; :: thesis: {|(the_hom_sets_of C),(the_hom_sets_of C)|} . e = {}
then ( Hom (i,j) = {} or Hom (j,k) = {} ) by A9, CAT_1:24;
hence {|(the_hom_sets_of C),(the_hom_sets_of C)|} . e = {} by A10, ZFMISC_1:90; :: thesis: verum

end;

the Comp of C .: [:(Hom (j,k)),(Hom (i,j)):] c= Hom (i,k) by Th12;
then A12: rng f c= {|(the_hom_sets_of C)|} . e by A8, A7, A9, RELAT_1:115;
[:(Hom (j,k)),(Hom (i,j)):] c= dom the Comp of C by Th11;
then dom f = {|(the_hom_sets_of C),(the_hom_sets_of C)|} . e by A8, A7, A10, RELAT_1:62;
hence o . e is Function of ({|(the_hom_sets_of C),(the_hom_sets_of C)|} . e),({|(the_hom_sets_of C)|} . e) by A11, A12, FUNCT_2:def 1, RELSET_1:4; :: thesis: verum

end;

then reconsider o = o as BinComp of (the_hom_sets_of C) by PBOOLE:def 15;
take o ; :: thesis: for i, j, k being Object of C holds o . (i,j,k) = the Comp of C | [:((the_hom_sets_of C) . (j,k)),((the_hom_sets_of C) . (i,j)):]
let i, j, k be Object of C; :: thesis: o . (i,j,k) = the Comp of C | [:((the_hom_sets_of C) . (j,k)),((the_hom_sets_of C) . (i,j)):]
reconsider e = [i,j,k] as Element of [: the carrier of C, the carrier of C, the carrier of C:] ;
A13: ([i,j,k] `1) `1 = e `1 by MCART_1:46
.= i by MCART_1:43 ;
A14: ([i,j,k] `1) `2 = e `2 by MCART_1:46
.= j by MCART_1:43 ;
A15: [i,j,k] `2 = e `3 by MCART_1:46
.= k by MCART_1:43 ;
thus o . (i,j,k) = o . [i,j,k] by MULTOP_1:def 1
.= the Comp of C | [:((the_hom_sets_of C) . (j,k)),((the_hom_sets_of C) . (i,j)):] by A2, A13, A14, A15 ; :: thesis: verum