
theorem Th1:
  for k,l,m being Nat holds (k(#)(l GeoSeq))|m is XFinSequence of NAT
proof
  let k,l,m be Nat;
  set g=(k(#)(l GeoSeq))|m;
A1: dom (k(#)(l GeoSeq)) = NAT by FUNCT_2:def 1;
  then m in dom (k(#)(l GeoSeq)) by ORDINAL1:def 12;
  then m c= dom (k(#)(l GeoSeq)) by A1,ORDINAL1:def 2;
  then dom g = m by RELAT_1:62;
  then reconsider g9=g as XFinSequence by ORDINAL1:def 7;
  rng g9 c= NAT
  proof
    let a be object;
    assume a in rng g9;
    then consider o being object such that
A2: o in dom g and
A3: a=g.o by FUNCT_1:def 3;
    o in dom (k(#)(l GeoSeq)) by A2,RELAT_1:57;
    then reconsider o as Element of NAT;
A4: k*(l|^o) in NAT by ORDINAL1:def 12;
    g.o = (k(#)(l GeoSeq)).o by A2,FUNCT_1:47
      .= k*((l GeoSeq).o) by SEQ_1:9
      .= k*(l|^o) by PREPOWER:def 1;
    hence thesis by A3,A4;
  end;
  hence thesis by RELAT_1:def 19;
end;
