reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;

theorem Th6:
  p in rng f1 implies p..(f1^f2) = p..f1
proof
A1: dom f1 c= dom(f1^f2) by FINSEQ_1:26;
  assume
A2: p in rng f1;
  then
A3: p..f1 in dom f1 by FINSEQ_4:20;
A4: now
A5: (f1^f2).(p..f1) = f1.(p..f1) by A3,FINSEQ_1:def 7;
    let i such that
A6: 1 <= i and
A7: i < p..f1;
    p..f1 <= len f1 by A2,FINSEQ_4:21;
    then i <= len f1 by A7,XXREAL_0:2;
    then
A8: i in dom f1 by A6,FINSEQ_3:25;
    then (f1^f2).i = f1.i by FINSEQ_1:def 7;
    hence (f1^f2).i <> (f1^f2).(p..f1) by A2,A7,A8,A5,FINSEQ_4:19,24;
  end;
  f1.(p..f1) = p by A2,FINSEQ_4:19;
  then (f1^f2).(p..f1) = p by A3,FINSEQ_1:def 7;
  hence thesis by A3,A1,A4,Th2;
end;
