reserve f for Function;
reserve n,k,n1 for Element of NAT;
reserve r,p for Complex;
reserve x,y for set;
reserve seq,seq1,seq2,seq3,seq9,seq19 for Complex_Sequence;

theorem Th2:
  f is Complex_Sequence iff (dom f=NAT & for n holds f.n is Element of COMPLEX)
proof
  thus f is Complex_Sequence implies (dom f=NAT & for n holds f.n is Element
  of COMPLEX) by Th1;
  assume that
A1: dom f=NAT and
A2: for n holds f.n is Element of COMPLEX;
  for x holds x in NAT implies f.x is Element of COMPLEX by A2;
  hence thesis by A1,Th1;
end;
