theorem Th59:
  dom(P * FSeq) = dom FSeq
proof
  for x being object holds x in dom (P * FSeq) iff x in dom FSeq
  proof
    let x be object;
    thus x in dom (P * FSeq) implies x in dom FSeq by FUNCT_1:11;
    assume
A1: x in dom FSeq;
    then reconsider k=x as Element of NAT;
    FSeq.k in Sigma;
    then FSeq.k in dom P by FUNCT_2:def 1;
    hence thesis by A1,FUNCT_1:11;
  end;
  hence thesis by TARSKI:2;
end;
