reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem
  f in doms F implies f|n in doms (F|n)
proof
  assume
A1: f in doms F;
  then
A2: len f = len F by Th47;
  per cases;
  suppose n > len f;
    then f|n=f & F|n=F by A2,FINSEQ_1:58;
    hence thesis by A1;
  end;
  suppose n <=len f;
    then
A3:   len (f|n)=n & len (F|n)=n by A2,FINSEQ_1:59;
    then
A4:   dom (f|n)=dom (F|n) by FINSEQ_3:30;
    for i st i in dom (f|n) holds (f|n).i in dom ((F|n).i)
    proof
      let i;
      assume i in dom (f|n);
      then (f|n).i = f.i & (F|n).i = F.i & i in dom f
        by A4,FUNCT_1:47,RELAT_1:57;
      hence thesis by A1,Th47;
    end;
    hence thesis by A3,Th47;
  end;
end;
