
theorem Th3:
  for x,y,z being set st <*x,y,z*> is constant holds x = y & y = z & z = x
proof
  let x,y,z be set;
A1: rng<*x,y,z*> = rng(<*x*>^<*y*>^<*z*>) by FINSEQ_1:def 10
    .= rng(<*x*>^<*y*>) \/ rng<*z*> by FINSEQ_1:31
    .= rng(<*x*>^<*y*>) \/ {z} by FINSEQ_1:38
    .= rng<*x*> \/ rng<*y*> \/ {z} by FINSEQ_1:31
    .= rng<*x*> \/ {y} \/ {z} by FINSEQ_1:38
    .= {x} \/ {y} \/ {z} by FINSEQ_1:38
    .= {x,y} \/ {z} by ENUMSET1:1
    .= {x,y,z} by ENUMSET1:3;
A2: y in {x,y,z} by ENUMSET1:def 1;
  assume <*x,y,z*> is constant;
  then reconsider s = <*x,y,z*> as constant Function;
A3: x in {x,y,z} by ENUMSET1:def 1;
A4: z in {x,y,z} by ENUMSET1:def 1;
  rng s is trivial;
  hence thesis by A1,A3,A2,A4,ZFMISC_1:def 10;
end;
