reserve f for Function;
reserve n,k,n1 for Nat;
reserve r,p for Real;
reserve x,y,z for object;
reserve seq,seq1,seq2,seq3,seq9,seq19 for Real_Sequence;

theorem Th1:
  f is Real_Sequence iff (dom f=NAT & for x st x in NAT holds f.x is real)
proof
  thus f is Real_Sequence implies (dom f=NAT & for x st x in NAT holds f.x is
  real) by FUNCT_2:def 1;
  assume that
A1: dom f= NAT and
A2: for x st x in NAT holds f.x is real;
  now
    let y be object;
    assume y in rng f;
    then consider x being object such that
A3: x in dom f and
A4: y=f.x by FUNCT_1:def 3;
    f.x is real by A1,A2,A3;
    hence y in REAL by A4,XREAL_0:def 1;
  end;
  then rng f c= REAL by TARSKI:def 3;
  hence thesis by A1,FUNCT_2:def 1,RELSET_1:4;
end;
