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

theorem
  <*x,y*>|--x = <*y*>
proof
A1: x..<*x,y*> = 1 by Th19;
  then len<*y*> + x..<*x,y*> = 1 + 1 by FINSEQ_1:40
    .= len<*x,y*> by FINSEQ_1:44;
  then
A2: len<*y*> = len<*x,y*> - x..<*x,y*>;
A3: now
    let k;
    assume k in dom<*y*>;
    then k in Seg 1 by FINSEQ_1:38;
    then
A4: k = 1 by FINSEQ_1:2,TARSKI:def 1;
    hence <*y*>.k = y
      .= <*x,y*>.(k + x..<*x,y*>) by A1,A4;
  end;
  x in { x,y } by TARSKI:def 2;
  then x in rng<*x,y*> by Lm1;
  hence thesis by A2,A3,FINSEQ_4:def 6;
end;
