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

theorem Th22:
  p1 <> p2 implies p2..<*p1,p2,p3*> = 2
proof
A2: <*p1,p2,p3*>.1 = p1;
  assume
A3: p1 <> p2;
A4: now
    let i;
    assume
A5: 1<=i;
    assume i<1+1;
    then i <= 1 by NAT_1:13;
    hence <*p1,p2,p3*>.i <> <*p1,p2,p3*>.2 by A3,A2,A5,XXREAL_0:1;
  end;
  len<*p1,p2,p3*> = 3 by FINSEQ_1:45;
  then 2 in dom<*p1,p2,p3*> by FINSEQ_3:25;
  hence thesis by A4,Th2;
end;
