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 Th11:
  rng seq c= dom (h^) implies (h^)/*seq =(h/*seq)"
proof
  assume
A1: rng seq c= dom (h^);
  then
A2: dom h \ h"{0c} c= dom h & rng seq c= dom h \ h"{0c} by CFUNCT_1:def 2
,XBOOLE_1:36;
  now
    let n be Element of NAT;
A3: seq.n in rng seq by VALUED_0:28;
    thus ((h^)/*seq).n = (h^)/.(seq.n) by A1,FUNCT_2:109
      .= (h/.(seq.n))" by A1,A3,CFUNCT_1:def 2
      .= ((h/*seq).n)" by A2,FUNCT_2:109,XBOOLE_1:1
      .= ((h/*seq)").n by VALUED_1:10;
  end;
  hence thesis by FUNCT_2:63;
end;
