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 Th51:
  p1 <> p3 & p2 <> p3 implies <*p1,p2,p3*>-:p3 = <*p1,p2,p3*>
proof
  assume that
A1: p1 <> p3 and
A2: p2 <> p3;
  rng<*p1,p2,p3*> = { p1,p2,p3 } by Lm2;
  then p3 in rng<*p1,p2,p3*> by ENUMSET1:def 1;
  hence <*p1,p2,p3*>-:p3 = (<*p1,p2,p3*> -| p3)^<*p3*> by Th40
    .= <*p1,p2*>^<*p3*> by A1,A2,Th28
    .= <*p1,p2,p3*>;
end;
