let f1, f2 be Function; :: thesis: ( dom f1 = the Sorts of A . ((the_arity_of o) /. i) & ( for x being set st x in the Sorts of A . ((the_arity_of o) /. i) holds
f1 . x = (Den o,A) . (a +* i,x) ) & dom f2 = the Sorts of A . ((the_arity_of o) /. i) & ( for x being set st x in the Sorts of A . ((the_arity_of o) /. i) holds
f2 . x = (Den o,A) . (a +* i,x) ) implies f1 = f2 )
assume that
A1: dom f1 = the Sorts of A . ((the_arity_of o) /. i) and
A2: for x being set st x in the Sorts of A . ((the_arity_of o) /. i) holds
f1 . x = (Den o,A) . (a +* i,x) and
A3: dom f2 = the Sorts of A . ((the_arity_of o) /. i) and
A4: for x being set st x in the Sorts of A . ((the_arity_of o) /. i) holds
f2 . x = (Den o,A) . (a +* i,x) ; :: thesis: f1 = f2

now

let x be set ; :: thesis: ( x in the Sorts of A . ((the_arity_of o) /. i) implies f1 . x = f2 . x )
assume x in the Sorts of A . ((the_arity_of o) /. i) ; :: thesis: f1 . x = f2 . x
then ( f1 . x = (Den o,A) . (a +* i,x) & f2 . x = (Den o,A) . (a +* i,x) ) by A2, A4;
hence f1 . x = f2 . x ; :: thesis: verum

end;

hence f1 = f2 by A1, A3, FUNCT_1:9; :: thesis: verum