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

theorem Th30:
  x <> y implies <*x,y,z*>|--y = <*z*>
proof
  assume x <> y;
  then
A1: y..<*x,y,z*> = 2 by Th22;
  then len<*z*> + y..<*x,y,z*> = 1 + 2 by FINSEQ_1:40
    .= len<*x,y,z*> by FINSEQ_1:45;
  then
A2: len<*z*> = len<*x,y,z*> - y..<*x,y,z*>;
A3: now
    let k;
    assume k in dom<*z*>;
    then k in Seg 1 by FINSEQ_1:38;
    then
A4: k = 1 by FINSEQ_1:2,TARSKI:def 1;
    hence <*z*>.k = z
      .= <*x,y,z*>.(k + y..<*x,y,z*>) by A1,A4;
  end;
  y in { x,y,z } by ENUMSET1:def 1;
  then y in rng<*x,y,z*> by Lm2;
  hence thesis by A2,A3,FINSEQ_4:def 6;
end;
