
theorem Th7:
  for D being non empty set for Q0,Q1 being FinSequence of D, i
  being Element of NAT st 1<=i & i<=len Q0 holds (Q0^Q1)/.i=Q0/.i
proof
  let D be non empty set;
  let Q0,Q1 be FinSequence of D, i be Element of NAT;
  len Q0<=len Q0+len Q1 by NAT_1:11;
  then
A1: i<=len Q0 implies i<=len Q0 + len Q1 by XXREAL_0:2;
  i in dom Q0 implies i in Seg(len Q0) by FINSEQ_1:def 3;
  then i in dom Q0 implies 1<=i & i<=len(Q0^Q1) by A1,FINSEQ_1:1,22;
  then
A2: i in dom Q0 implies i in dom (Q0^Q1) by FINSEQ_3:25;
  i in dom Q0 implies Q0.i=Q0/.i by PARTFUN1:def 6;
  then
A3: i in dom Q0 implies (Q0^Q1).i=Q0/.i by FINSEQ_1:def 7;
  i in dom Q0 iff i in Seg len Q0 by FINSEQ_1:def 3;
  hence thesis by A2,A3,FINSEQ_1:1,PARTFUN1:def 6;
end;
