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 Th69:
  p1 in rng f & p2 in rng f \ rng(f-:p1) implies f|--p2 = f|--p1 |--p2
proof
  assume that
A1: p1 in rng f and
A2: p2 in rng f \ rng(f-:p1);
  not p2 in rng(f-:p1) by A2,XBOOLE_0:def 5;
  then
A3: not p2 in rng((f-|p1)^<* p1 *>) by A1,Th40;
  f = (f-|p1)^<* p1 *>^(f|--p1) by A1,FINSEQ_4:51;
  then rng f = rng((f-|p1)^<* p1 *>) \/ rng(f|--p1) by FINSEQ_1:31;
  then p2 in rng(f|--p1) by A2,A3,XBOOLE_0:def 3;
  then
A4: p2 in rng(f|--p1) \ rng((f-|p1)^<* p1 *>) by A3,XBOOLE_0:def 5;
  thus f|--p2 = ((f-|p1)^<* p1 *>^(f|--p1))|--p2 by A1,FINSEQ_4:51
    .= f|--p1|--p2 by A4,Th9;
end;
