reconsider D = dom F as finite set by A1;
let n1, n2 be Element of NAT ; :: thesis:
assume that
A61: for x, y being set
for X being finite set
for P being Function of (card (Choose X,k,x,y)),(Choose X,k,x,y) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of NAT st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection F,f,x) ) & n1 = Sum XFS ) and
A62: for x, y being set
for X being finite set
for P being Function of (card (Choose X,k,x,y)),(Choose X,k,x,y) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of NAT st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection F,f,x) ) & n2 = Sum XFS ) ; :: thesis:
set Ch1 = Choose D,k,0 ,1;
card (Choose D,k,0 ,1), Choose D,k,0 ,1 are_equipotent by CARD_1:def 5;
then consider P being Function such that
A63: P is one-to-one and
A64: ( dom P = card (Choose D,k,0 ,1) & rng P = Choose D,k,0 ,1 ) by WELLORD2:def 4;
reconsider P = P as Function of (card (Choose D,k,0 ,1)),(Choose D,k,0 ,1) by A64, FUNCT_2:3;
consider XFS1 being XFinSequence of NAT such that
A65: dom XFS1 = dom P and
A66: for z being set
for f being Function st z in dom XFS1 & f = P . z holds
XFS1 . z = card (Intersection F,f,0 ) and
A67: n1 = Sum XFS1 by A61, A63;
consider XFS2 being XFinSequence of NAT such that
A68: dom XFS2 = dom P and
A69: for z being set
for f being Function st z in dom XFS2 & f = P . z holds
XFS2 . z = card (Intersection F,f,0 ) and
A70: n2 = Sum XFS2 by A62, A63;

now

let z be set ; :: thesis:
assume A71: z in dom XFS1 ; :: thesis:
P . z in rng P by A65, A71, FUNCT_1:def 5;
then consider Pz being Function of D,{0 ,1} such that
A72: Pz = P . z and
card (Pz " {0 }) = k by Def1;
XFS2 . z = card (Intersection F,Pz,0 ) by A65, A68, A69, A71, A72;
hence XFS2 . z = XFS1 . z by A66, A71, A72; :: thesis:

end;

hence n1 = n2 by A65, A67, A68, A70, FUNCT_1:9; :: thesis: