reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;

theorem
  (ex k st dom f c= k) implies ex p st f c= p
proof
  given k such that
A1: dom f c= k;
  deffunc F(object) = f.$1;
  consider g such that
A2: dom g = k &
for x being object st x in k holds g.x = F(x) from FUNCT_1:sch 3;
  reconsider g as XFinSequence by A2,FINSET_1:10,ORDINAL1:def 7;
  take g;
  let y,z be object;
  assume A3: [y,z] in f;
  then
A4: y in dom f by XTUPLE_0:def 12;
  then
A5: [y,g.y] in g by A1,A2,FUNCT_1:1;
  z is set by TARSKI:1;
  then f.y = z by A3,A4,FUNCT_1:def 2;
  hence thesis by A1,A2,A4,A5;
end;
