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,z*>|--x = <*y,z*>
proof
A1: x..<*x,y,z*> = 1 by Th21;
  then len<*y,z*> + x..<*x,y,z*> = 2 + 1 by FINSEQ_1:44
    .= len<*x,y,z*> by FINSEQ_1:45;
  then
A2: len<*y,z*> = len<*x,y,z*> - x..<*x,y,z*>;
A3: len<*y,z*> = 2 by FINSEQ_1:44;
A4: now
    let k;
    assume k in dom<*y,z*>;
    then
A5: k in Seg 2 by A3,FINSEQ_1:def 3;
    per cases by A5,FINSEQ_1:2,TARSKI:def 2;
    suppose
A6:   k = 1;
      hence <*y,z*>.k = y
        .= <*x,y,z*>.(k + x..<*x,y,z*>) by A1,A6;
    end;
    suppose
A7:   k = 2;
      hence <*y,z*>.k = z
        .= <*x,y,z*>.(k + x..<*x,y,z*>) by A1,A7;
    end;
  end;
  x in { x,y,z } by ENUMSET1:def 1;
  then x in rng<*x,y,z*> by Lm2;
  hence thesis by A2,A4,FINSEQ_4:def 6;
end;
