reserve i,j,k,n for Nat;
reserve D for non empty set,
  p for Element of D,
  f,g for FinSequence of D;

theorem
  dom f = dom g & (for i st i in dom f holds f/.i = g/.i) implies f = g
proof
  assume that
A1: dom f = dom g and
A2: for i st i in dom f holds f/.i = g/.i;
  now
    let k be Nat;
    assume
A3: k in dom f;
    hence f.k = f/.k by PARTFUN1:def 6
      .= g/.k by A2,A3
      .= g.k by A1,A3,PARTFUN1:def 6;
  end;
  hence thesis by A1;
end;
