reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;

theorem Th23:
  p1 <> p3 & p2 <> p3 implies p3..<*p1,p2,p3*> = 3
proof
  assume that
A1: p1 <> p3 and
A2: p2 <> p3;
A6: now
    let i;
    assume 1<=i;
    then
A7: i <> 0;
    assume i<2+1;
    then i <= 2 by NAT_1:13;
    then i = 0 or ... or i = 2;
    hence <*p1,p2,p3*>.i <> <*p1,p2,p3*>.3 by A1,A2,A7;
  end;
  3 <= len<*p1,p2,p3*> by FINSEQ_1:45;
  then 3 in dom<*p1,p2,p3*> by FINSEQ_3:25;
  hence thesis by A6,Th2;
end;
