let C be para-functional category; for a, b being Object of C st <^a,b^> <> {} holds
for f being Morphism of a,b holds
( dom f = the_carrier_of a & rng f c= the_carrier_of b )
let a, b be Object of C; ( <^a,b^> <> {} implies for f being Morphism of a,b holds
( dom f = the_carrier_of a & rng f c= the_carrier_of b ) )
assume A1:
<^a,b^> <> {}
; for f being Morphism of a,b holds
( dom f = the_carrier_of a & rng f c= the_carrier_of b )
let f be Morphism of a,b; ( dom f = the_carrier_of a & rng f c= the_carrier_of b )
A2:
<^a,b^> c= Funcs ((the_carrier_of a),(the_carrier_of b))
by Th33;
f in <^a,b^>
by A1;
hence
( dom f = the_carrier_of a & rng f c= the_carrier_of b )
by A2, FUNCT_2:92; verum