reconsider n = n as Nat by TARSKI:1;
Seg n is finite

proof

A1: n = { k where k is Nat : k < n } by AXIOMS:4;
A2: for x being object st x in n holds
ex y being object st S₁[x,y]

proof

let x be object ; :: thesis: ( x in n implies ex y being object st S₁[x,y] )
assume x in n ; :: thesis: ex y being object st S₁[x,y]
then ex k being Nat st
( x = k & k < n ) by A1;
then reconsider k = x as Nat ;
take k + 1 ; :: thesis: S₁[x,k + 1]
thus S₁[x,k + 1] ; :: thesis: verum

end;

consider f being Function such that
A3: dom f = n and
A4: for x being object st x in n holds
S₁[x,f . x] from CLASSES1:sch 1(A2);
take f ; :: according to FINSET_1:def 1 :: thesis: ( rng f = Seg n & dom f in omega )
thus rng f = Seg n :: thesis: dom f in omega

proof

thus rng f c= Seg n :: according to XBOOLE_0:def 10 :: thesis: Seg n c= rng f

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng f or x in Seg n )
assume x in rng f ; :: thesis: x in Seg n
then consider y being object such that
A5: y in dom f and
A6: x = f . y by FUNCT_1:def 3;
consider k being Nat such that
A7: y = k and
A8: x = k + 1 by A3, A4, A5, A6;
ex m being Nat st
( m = y & m < n ) by A1, A3, A5;
then ( 1 <= k + 1 & k + 1 <= n ) by A7, NAT_1:11, NAT_1:13;
hence x in Seg n by A8; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Seg n or x in rng f )
assume x in Seg n ; :: thesis: x in rng f
then consider k being Nat such that
A9: x = k and
A10: 1 <= k and
A11: k <= n ;
consider i being Nat such that
A12: 1 + i = k by A10, NAT_1:10;
( i in NAT & i < n ) by A11, A12, NAT_1:13, ORDINAL1:def 12;
then A13: i in n by A1;
then S₁[i,f . i] by A4;
hence x in rng f by A3, A9, A12, A13, FUNCT_1:def 3; :: thesis: verum

end;

thus dom f in omega by A3, ORDINAL1:def 12; :: thesis: verum

end;

hence Seg n is finite ; :: thesis: verum