reserve
  a for natural Number,
  k,l,m,n,k1,b,c,i for Nat,
  x,y,z,y1,y2 for object,
  X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for FinSequence;

theorem
  (ex k st dom f c= Seg k) implies ex p st f c= p
proof
  given k such that
A1: dom f c= Seg k;
  deffunc F(object) = f.$1;
  consider g such that
A2: dom g = Seg k &
for x being object st x in Seg k holds g.x = F(x) from FUNCT_1:sch 3;
  reconsider g as FinSequence by A2,Def2;
  take g;
  let y,z be object;
  assume
A3: [y,z] in f; then
A4: y in dom f by XTUPLE_0:def 12;
  reconsider z as set by TARSKI:1;
A5: f.y = z by A3,FUNCT_1:def 2,A4;
  [y,g.y] in g by A1,A2,A4,FUNCT_1:1;
  hence thesis by A1,A2,A4,A5;
end;
