reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;
reserve f for Function of A,B;

theorem
  for m being Nat for D being set, p being FinSequence of D st
    a in dom(p|m) holds (p|m qua FinSequence of D)/.a = p/.a
proof
  let m be Nat;
  let D be set, p be FinSequence of D;
  assume
A1: a in dom(p|m);
  then a in dom p /\ (Seg m) by RELAT_1:61;
  then
A2: a in dom p by XBOOLE_0:def 4;
  thus (p|m)/.a = (p|Seg m).a by A1,PARTFUN1:def 6
    .= p.a by A1,FUNCT_1:47
    .= p/.a by A2,PARTFUN1:def 6;
end;
