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 Th78:
  p2 in rng f & p2..f <> 1 implies p2 in rng(f/^1)
proof
  assume that
A1: p2 in rng f and
A2: p2..f <> 1;
  f = <*f/.1*>^(f/^1) by A1,FINSEQ_5:29,RELAT_1:38;
  then
A3: rng f = rng<*f/.1*> \/ rng(f/^1) by FINSEQ_1:31;
  assume not p2 in rng(f/^1);
  then p2 in rng<*f/.1*> by A1,A3,XBOOLE_0:def 3;
  then p2 in { f/.1 } by FINSEQ_1:39;
  then p2 = f/.1 by TARSKI:def 1;
  hence contradiction by A1,A2,Th43,RELAT_1:38;
end;
