reserve n,n1,m,m1,k for Nat;
reserve x,X,X1 for set;
reserve g,g1,g2,t,x0,x1,x2 for Complex;
reserve s1,s2,q1,seq,seq1,seq2,seq3 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f,f1,f2,h,h1,h2 for PartFunc of COMPLEX,COMPLEX;
reserve p,r,s for Real;
reserve Ns,Nseq for increasing sequence of NAT;

theorem
  rng seq c= dom h implies Im ( (h/*seq)*Ns ) = Im (h/*(seq*Ns))
proof
  assume
A1: rng seq c= dom h;
  now
    let n be Element of NAT;
    (seq * Ns) is subsequence of seq by VALUED_0:def 17;
    then
A2: rng (seq*Ns) c= rng seq by VALUED_0:21;
    thus (Im ( (h/*seq)*Ns )).n = Im( ((h/*seq)*Ns).n ) by COMSEQ_3:def 6
      .= Im( (h/*seq).(Ns.n) ) by FUNCT_2:15
      .= Im( h/.(seq.(Ns.n)) ) by A1,FUNCT_2:109
      .=Im( h/.((seq*Ns).n) ) by FUNCT_2:15
      .=Im( (h/*(seq*Ns)).n ) by A1,A2,FUNCT_2:109,XBOOLE_1:1
      .=(Im (h/*(seq*Ns)) ).n by COMSEQ_3:def 6;
  end;
  hence thesis by FUNCT_2:63;
end;
