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 Th25:
  for a1,a2,a3,a4,a5,a6,a7,a8,a9,a10 being object holds
  dom (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>) = Seg 10 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).1 = a1 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).2 = a2 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).3 = a3 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).4 = a4 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).5 = a5 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).6 = a6 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).7 = a7 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).8 = a8 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).9 = a9 &
  (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).10 = a10
  proof
    let a1,a2,a3,a4,a5,a6,a7,a8,a9,a10 be object;
    thus dom (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>)
    = Seg (len <*a1,a2,a3,a4,a5,a6,a7,a8*> + len <*a9,a10*>) by FINSEQ_1:def 7
    .= Seg (8+len <*a9,a10*>) by Th19
    .= Seg (8+2) by FINSEQ_1:44 .= Seg 10;
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,a10*>).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,a10*>).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,a10*>).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,a10*>).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,a10*>).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,a10*>).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,a10*>).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,a10*>).8
    = <*a1,a2,a3,a4,a5,a6,a7,a8*>.8 by FINSEQ_1:def 7 .= a8;
    len <*a9,a10*> = 2 by FINSEQ_1:44; then
    1 in dom <*a9,a10*> & 9 = 8+1 by FINSEQ_3:25;
    hence (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).9
    = <*a9,a10*>.1 by A1,FINSEQ_1:def 7 .= a9;
    len <*a9,a10*> = 2 by FINSEQ_1:44; then
    2 in dom <*a9,a10*> by FINSEQ_3:25;
    hence (<*a1,a2,a3,a4,a5,a6,a7,a8*>^<*a9,a10*>).10
    = <*a9,a10*>.2 by A1,FINSEQ_1:def 7 .= a10;
  end;
