reserve D for non empty set;
reserve s for FinSequence of D;
reserve m,n for Element of NAT;

theorem Th29:
  for r be Element of 48-tuples_on BOOLEAN
  ex s1,s2,s3,s4,s5,s6,s7,s8
  be Element of 6-tuples_on BOOLEAN st
  s1=(DES-DIV8(r)).1&
  s2=(DES-DIV8(r)).2&
  s3=(DES-DIV8(r)).3&
  s4=(DES-DIV8(r)).4&
  s5=(DES-DIV8(r)).5&
  s6=(DES-DIV8(r)).6&
  s7=(DES-DIV8(r)).7&
  s8=(DES-DIV8(r)).8&
  r=s1^s2^s3^s4^s5^s6^s7^s8
  proof
    let r be Element of 48-tuples_on BOOLEAN;
    r is Element of Funcs(Seg 48,BOOLEAN) by FINSEQ_2:93;
    then
    dom r =Seg 48 by FUNCT_2:def 1;then
    A1: len r =48 by FINSEQ_1:def 3;
    1 in Seg 8;then
    reconsider SEG8E1=1 as Element of Seg 8;
    2 in Seg 8;then
    reconsider SEG8E2=2 as Element of Seg 8;
    3 in Seg 8;then
    reconsider SEG8E3=3 as Element of Seg 8;
    4 in Seg 8;then
    reconsider SEG8E4=4 as Element of Seg 8;
    5 in Seg 8;then
    reconsider SEG8E5=5 as Element of Seg 8;
    6 in Seg 8;then
    reconsider SEG8E6=6 as Element of Seg 8;
    7 in Seg 8;then
    reconsider SEG8E7=7 as Element of Seg 8;
    8 in Seg 8;then
    reconsider SEG8E8=8 as Element of Seg 8;

    set s1=(DES-DIV8(r)).SEG8E1,
    s2=(DES-DIV8(r)).SEG8E2,
    s3=(DES-DIV8(r)).SEG8E3,
    s4=(DES-DIV8(r)).SEG8E4,
    s5=(DES-DIV8(r)).SEG8E5,
    s6=(DES-DIV8(r)).SEG8E6,
    s7=(DES-DIV8(r)).SEG8E7,
    s8=(DES-DIV8(r)).SEG8E8;

    take s1,s2,s3,s4,s5,s6,s7,s8;

    r=s1^s2^s3^s4^s5^s6^s7^s8
    proof
      A2:
      s1 =Op-Left(r,6) &
      s2 =Op-Left(Op-Right(r,6),6) &
      s3 =Op-Left(Op-Right(r,12),6) &
      s4 =Op-Left(Op-Right(r,18),6) &
      s5 =Op-Left(Op-Right(r,24),6) &
      s6 =Op-Left(Op-Right(r,30),6) &
      s7 =Op-Left(Op-Right(r,36),6) &
      s8 =Op-Right(r,42) by Def20;

      s1^s2=r|12 by Th18,A1,A2;then
      s1^s2^s3
      =r|18 by Th18,A1,A2;then
      s1^s2^s3^s4
      =r|24 by Th18,A1,A2;then
      s1^s2^s3^s4^s5
      =r|30 by Th18,A1,A2;then
      s1^s2^s3^s4^s5^s6
      =r|36 by Th18,A1,A2;then
      s1^s2^s3^s4^s5^s6^s7
      =r|42 by Th18,A1,A2;
      hence thesis by A2,RFINSEQ:8;
    end;
    hence thesis;
  end;
