let f be Function; :: thesis: ( ex k being Nat st dom f c= k implies ex p being XFinSequence st f c= p )
given k being Nat such that A1: dom f c= k ; :: thesis: ex p being XFinSequence st f c= p
deffunc H₁( set ) -> set = f . $1;
consider g being Function such that
A2: ( dom g = k & ( for x being set st x in k holds
g . x = H₁(x) ) ) from FUNCT_1:sch 3();
reconsider g = g as XFinSequence by A2, Th7;
take g ; :: thesis: f c= g
let y be set ; :: according to RELAT_1:def 3 :: thesis: for b₁ being set holds
( not [y,b₁] in f or [y,b₁] in g )
let z be set ; :: thesis: ( not [y,z] in f or [y,z] in g )
assume A3: [y,z] in f ; :: thesis: [y,z] in g
then A4: y in dom f by RELAT_1:def 4;
then A5: [y,(g . y)] in g by A1, A2, FUNCT_1:1;
f . y = z by A3, A4, FUNCT_1:def 2;
hence [y,z] in g by A1, A2, A4, A5; :: thesis: verum