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

theorem Th67:
  p in rng f2 \ rng f1 implies (f1^f2)-:p = f1^(f2-:p)
proof
  assume
A1: p in rng f2 \ rng f1;
  rng(f1^f2) = rng f1 \/ rng f2 by FINSEQ_1:31;
  then p in rng(f1^f2) by A1,XBOOLE_0:def 3;
  hence (f1^f2)-:p = ((f1^f2)-|p)^<*p*> by Th40
    .= f1^(f2-|p)^<*p*> by A1,Th15
    .= f1^((f2-|p)^<*p*>) by FINSEQ_1:32
    .= f1^(f2-:p) by A1,Th40;
end;
