
theorem Th1:
  for a1,a2,a3,a4,a5,a6,a7,a8,a9 being object
  for f being FinSequence holds f = <*a1,a2,a3,a4,a5,a6,a7,a8,a9*> iff
  len f = 9 &
  f.1 = a1 & f.2 = a2 & f.3 = a3 & f.4 = a4 & f.5 = a5 & f.6 = a6 & f.7 = a7 &
  f.8 = a8 & f.9 = a9
  proof
    let a1,a2,a3,a4,a5,a6,a7,a8,a9 be object;
    let f be FinSequence;
    set AS1 = <*a9*>;
    set AS8 = <*a1,a2,a3,a4,a5,a6,a7,a8*>;
    set AS9 = <*a1,a2,a3,a4,a5,a6,a7,a8,a9*>;
A1: now
      let f be FinSequence;
      assume
A2:   f = AS9;
      hence len f = len AS8+ len AS1 by FINSEQ_1:22
      .= 8 + len AS1 by AOFA_A00:24
      .= 8 + 1 by FINSEQ_1:39
      .= 9;
      dom AS8 = Seg 8 by FINSEQ_1:89;
      then
      1 in dom AS8 & 2 in dom AS8 & 3 in dom AS8 & 4 in dom AS8 &
      5 in dom AS8 & 6 in dom AS8 & 7 in dom AS8 & 8 in dom AS8;
      then f.1 = AS8.1 & f.2 = AS8.2 & f.3 = AS8.3 & f.4 = AS8.4 &
      f.5 = AS8.5 & f.6 = AS8.6 & f.7 = AS8.7 & f.8 = AS8.8
      by A2, FINSEQ_1:def 7;
      hence f.1 = a1 & f.2 = a2 & f.3 = a3 & f.4 = a4 & f.5 = a5 &
      f.6 = a6 & f.7 = a7 & f.8 = a8;
      len AS8 = 8 by AOFA_A00:24;
      hence f.9 = a9 by A2;
    end;
    hence f = AS9 implies len f = 9 &
    f.1 = a1 & f.2 = a2 & f.3 = a3 & f.4 = a4 & f.5 = a5 & f.6 = a6 &
    f.7 = a7 & f.8 = a8 & f.9 = a9;
    assume
A3: len f = 9;
    len AS9 = 9 by A1;
    then
A4: dom f = Seg 9 & dom AS9 = Seg 9 by A3,FINSEQ_1:def 3;
    assume
A5: f.1 = a1 & f.2 = a2 & f.3 = a3 & f.4 = a4 & f.5 = a5 & f.6 = a6
    & f.7 = a7 & f.8 = a8 & f.9 = a9;
    now let x be object;
      assume x in Seg 9; then
      x = 1 or x = 2 or x = 3 or x = 4 or x = 5 or
      x = 6 or x = 7 or x = 8 or x = 9 by AOFA_A00:27,ENUMSET1:def 7;
      hence f.x = AS9.x by A1,A5;
    end;
    hence f = AS9 by A4, FUNCT_1:2;
  end;
