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

theorem Th15:
  p in rng f2 \ rng f1 implies (f1^f2)-|p = f1^(f2-|p)
proof
  assume
A1: p in rng f2 \ rng f1;
  then consider n being Nat such that
A2: n = p..f2 - 1 and
A3: f2 | Seg n = f2 -| p by FINSEQ_4:def 5;
  p..(f1^f2) = len f1 + p..f2 by A1,Th7;
  then
A4: len f1 + n = p..(f1^f2) - 1 by A2;
  rng(f1^f2) = rng f1 \/ rng f2 by FINSEQ_1:31;
  then
A5: p in rng(f1^f2) by A1,XBOOLE_0:def 3;
  (f1^f2) | Seg(len f1 + n) = f1^(f2-|p) by A3,Th14;
  hence thesis by A5,A4,FINSEQ_4:def 5;
end;
