reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th24:
  for a1,a2,a3,a4,a5,a6,a7,a8,a9 being object holds
  dom (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>) = Seg 9 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).1 = a1 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).2 = a2 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).3 = a3 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).4 = a4 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).5 = a5 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).6 = a6 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).7 = a7 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).8 = a8 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).9 = a9
  proof
    let a1,a2,a3,a4,a5,a6,a7,a8,a9 be object;
    thus dom (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>)
    = Seg (len <*a1,a2,a3,a4,a5,a6,a7,a8*> + len <*a9*>) by FINSEQ_1:def 7
    .= Seg (8+len <*a9*>) by Th19
    .= Seg (8+1) by FINSEQ_1:40 .= Seg 9;
A1: len <*a1,a2,a3,a4,a5,a6,a7,a8*> = 8 by Th19; then
    1 in dom <*a1,a2,a3,a4,a5,a6,a7,a8*> by FINSEQ_3:25;
    hence (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).1
    = <*a1,a2,a3,a4,a5,a6,a7,a8*>.1 by FINSEQ_1:def 7 .= a1;
    2 in dom <*a1,a2,a3,a4,a5,a6,a7,a8*> by A1,FINSEQ_3:25;
    hence (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).2
    = <*a1,a2,a3,a4,a5,a6,a7,a8*>.2 by FINSEQ_1:def 7 .= a2;
    3 in dom <*a1,a2,a3,a4,a5,a6,a7,a8*> by A1,FINSEQ_3:25;
    hence (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).3
    = <*a1,a2,a3,a4,a5,a6,a7,a8*>.3 by FINSEQ_1:def 7 .= a3;
    4 in dom <*a1,a2,a3,a4,a5,a6,a7,a8*> by A1,FINSEQ_3:25;
    hence (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).4
    = <*a1,a2,a3,a4,a5,a6,a7,a8*>.4 by FINSEQ_1:def 7 .= a4;
    5 in dom <*a1,a2,a3,a4,a5,a6,a7,a8*> by A1,FINSEQ_3:25;
    hence (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).5
    = <*a1,a2,a3,a4,a5,a6,a7,a8*>.5 by FINSEQ_1:def 7 .= a5;
    6 in dom <*a1,a2,a3,a4,a5,a6,a7,a8*> by A1,FINSEQ_3:25;
    hence (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).6
   = <*a1,a2,a3,a4,a5,a6,a7,a8*>.6 by FINSEQ_1:def 7 .= a6;
    7 in dom <*a1,a2,a3,a4,a5,a6,a7,a8*> by A1,FINSEQ_3:25;
    hence (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).7
    = <*a1,a2,a3,a4,a5,a6,a7,a8*>.7 by FINSEQ_1:def 7 .= a7;
    8 in dom <*a1,a2,a3,a4,a5,a6,a7,a8*> by A1,FINSEQ_3:25;
    hence (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).8
    = <*a1,a2,a3,a4,a5,a6,a7,a8*>.8 by FINSEQ_1:def 7 .= a8;
    len <*a9*> = 1 by FINSEQ_1:40; then
    1 in dom <*a9*> by FINSEQ_3:25;
    hence (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9*>).9
    = <*a9*>.1 by A1,FINSEQ_1:def 7 .= a9;
  end;
