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 Th55:
  p1 <> p3 & p2 <> p3 implies <*p1,p2,p3*>:-p3 = <*p3*>
proof
  assume that
A1: p1 <> p3 and
A2: p2 <> p3;
  p3 in { p1,p2,p3 } by ENUMSET1:def 1;
  then p3 in rng<*p1,p2,p3*> by Lm2;
  hence <*p1,p2,p3*>:-p3 = <*p3*>^(<*p1,p2,p3*> |-- p3) by Th41
    .= <*p3*>^{} by A1,A2,Th34
    .= <*p3*> by FINSEQ_1:34;
end;
