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 Th75:
  p1 in rng f & p2 in rng(f-:p1) implies f-:p1-:p2 = f-:p2
proof
  assume that
A1: p1 in rng f and
A2: p2 in rng(f-:p1);
  per cases;
  suppose
    p1 = p2;
    hence thesis by A1,Th74;
  end;
  suppose
    p1 <> p2;
    then not p2 in { p1 } by TARSKI:def 1;
    then
A3: not p2 in rng<*p1*> by FINSEQ_1:39;
    f-:p1 = (f-|p1)^<*p1*> by A1,Th40;
    then rng(f-:p1) = rng(f-|p1) \/ rng<*p1*> by FINSEQ_1:31;
    then
A4: p2 in rng(f-|p1) by A2,A3,XBOOLE_0:def 3;
A5: rng(f-|p1) c= rng f by A1,FINSEQ_4:39;
    thus f-:p1-:p2 = ((f-:p1)-|p2)^<*p2*> by A2,Th40
      .= (((f-|p1)^<*p1*>)-|p2)^<*p2*> by A1,Th40
      .= ((f-|p1)-|p2)^<*p2*> by A4,Th12
      .= (f-|p2)^<*p2*> by A1,A4,Th35
      .= f-:p2 by A4,A5,Th40;
  end;
end;
