
theorem Th3:
  for a1,a2,a3,a4,a5,a6,a7,a8,a9,a10 being object
  for f being FinSequence holds f = <*a1,a2,a3,a4,a5,a6,a7,a8,a9,a10*> iff
  len f = 10 &
  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 & f.10 = a10
  proof
    let a1,a2,a3,a4,a5,a6,a7,a8,a9,a10 be object;
    let f be FinSequence;
    set AS1 = <*a10*>;
    set AS9 = <*a1,a2,a3,a4,a5,a6,a7,a8,a9*>;
    set AS10 = <*a1,a2,a3,a4,a5,a6,a7,a8,a9,a10*>;
A1: now
      let f be FinSequence;
      assume
A2:   f = AS10;
      hence len f = len AS9+ len AS1 by FINSEQ_1:22
      .= 9 + len AS1 by Th1
      .= 9 + 1 by FINSEQ_1:39
      .= 10;
      dom AS9 = Seg 9 by FINSEQ_1:89;
      then
      1 in dom AS9 & 2 in dom AS9 & 3 in dom AS9 & 4 in dom AS9 &
      5 in dom AS9 & 6 in dom AS9 & 7 in dom AS9 & 8 in dom AS9 & 9 in dom AS9;
      then f.1 = AS9.1 & f.2 = AS9.2 & f.3 = AS9.3 & f.4 = AS9.4 &
      f.5 = AS9.5 & f.6 = AS9.6 & f.7 = AS9.7 & f.8 = AS9.8 & f.9 = AS9.9
      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 & f.9 = a9;
      len AS9 = 9 by Th1;
      hence f.10 = a10 by A2;
    end;
    hence f = AS10 implies len f = 10 &
    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 & f.10 = a10;
    assume
A3: len f = 10;
    len AS10 = 10 by A1;
    then
A4: dom f = Seg 10 & dom AS10 = Seg 10 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 & f.10 = a10;
    now let x be object;
      assume x in Seg 10; 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 or x = 10
      by AOFA_A00:28,ENUMSET1:def 8;
      hence f.x = AS10.x by A1,A5;
    end;
    hence f = AS10 by A4, FUNCT_1:2;
  end;
