theorem
  [#](p,d)|(dom p) = p
proof
  set k = len p, f = [#](p,d);
  Seg k c= NAT;
  then Seg k c= dom f by FUNCT_2:def 1;
  then
A1: dom (f|Seg k) = Seg k by RELAT_1:62;
A2: dom p = Seg k by FINSEQ_1:def 3;
  now
    let x be object;
    assume
A3: x in Seg k;
    then (f|Seg k).x = f.x by A1,FUNCT_1:47;
    hence (f|Seg k).x = p.x by A2,A3,Th20;
  end;
  hence thesis by A1,A2,FUNCT_1:2;
end;
