reserve i,j,k,x for object;

theorem Th10:
  for C being pseudo-functional non empty AltCatStr for a1,a2,a3
being Object of C st <^a1,a2^> <> {} & <^a2,a3^> <> {} & <^a1,a3^> <> {} for f
being Morphism of a1,a2, g being Morphism of a2,a3, f9,g9 being Function st f =
  f9 & g = g9 holds g*f =g9*f9
proof
  let C be pseudo-functional non empty AltCatStr;
  let a1,a2,a3 be Object of C such that
A1: <^a1,a2^> <> {} & <^a2,a3^> <> {} and
A2: <^a1,a3^> <> {};
  let f be Morphism of a1,a2, g be Morphism of a2,a3, f9,g9 be Function such
  that
A3: f = f9 & g = g9;
A4: [g,f] in [:<^a2,a3^>,<^a1,a2^>:] by A1,ZFMISC_1:87;
A5: (the Comp of C).(a1,a2,a3) = FuncComp(Funcs(a1,a2),Funcs(a2,a3))|([:<^a2
  ,a3^>,<^a1,a2^>:] qua set) by Def13;
  dom(FuncComp(Funcs(a1,a2),Funcs(a2,a3))|([:<^a2,a3^>,<^a1,a2^>:] qua set
)) c= dom(FuncComp(Funcs(a1,a2),Funcs(a2,a3))) & dom((the Comp of C).(a1,a2,a3)
  ) = [:<^a2,a3^>,<^a1,a2^>:] by A2,FUNCT_2:def 1,RELAT_1:60;
  then consider f99,g99 being Function such that
A6: [g,f] = [g99,f99] and
A7: FuncComp(Funcs(a1,a2),Funcs(a2,a3)).[g,f] = g99*f99 by A5,A4,Def9;
A8: g = g99 & f = f99 by A6,XTUPLE_0:1;
  thus g*f = (the Comp of C).(a1,a2,a3).(g,f) by A1,Def8
    .= g9*f9 by A5,A3,A4,A7,A8,FUNCT_1:49;
end;
