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 Th1:
  f is Complex_Sequence iff (dom f=NAT & for x st x in NAT holds f.
  x is Element of COMPLEX)
proof
  thus f is Complex_Sequence implies (dom f=NAT & for x st x in NAT holds f.x
  is Element of COMPLEX)
  proof
    assume
A1: f is Complex_Sequence;
    hence
A2: dom f=NAT by FUNCT_2:def 1;
    let x;
    assume x in NAT;
    then
A3: f.x in rng f by A2,FUNCT_1:def 3;
    rng f c=COMPLEX by A1,RELAT_1:def 19;
    hence thesis by A3;
  end;
  assume that
A4: dom f= NAT and
A5: for x st x in NAT holds f.x is Element of COMPLEX;
  now
    let y be object;
    assume y in rng f;
    then consider x being object such that
A6: x in dom f and
A7: y=f.x by FUNCT_1:def 3;
    f.x is Element of COMPLEX by A4,A5,A6;
    hence y in COMPLEX by A7;
  end;
  then rng f c=COMPLEX by TARSKI:def 3;
  hence thesis by A4,FUNCT_2:def 1,RELSET_1:4;
end;
