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

theorem Th22:
  for S be non empty set,
  x1,x2,x3,x4,x5,x6,x7,x8,
  x9,x10,x11,x12,x13,x14,x15,x16,
  x17,x18,x19,x20,x21,x22,x23,x24,
  x25,x26,x27,x28,x29,x30,x31,x32
  be Element of S holds
  ex s be FinSequence of S st
  s is 32-element &
  s.1=x1&s.2=x2&s.3=x3&s.4=x4&
  s.5=x5&s.6=x6&s.7=x7&s.8=x8&
  s.9=x9&s.10=x10&s.11=x11&s.12=x12&
  s.13=x13&s.14=x14&s.15=x15&s.16=x16 &
  s.17=x17&s.18=x18&s.19=x19&s.20=x20&
  s.21=x21&s.22=x22&s.23=x23&s.24=x24&
  s.25=x25&s.26=x26&s.27=x27&s.28=x28&
  s.29=x29&s.30=x30&s.31=x31&s.32=x32
  proof
    let S be non empty set,
    x1,x2,x3,x4,x5,x6,x7,x8,
    x9,x10,x11,x12,x13,x14,x15,x16,
    x17,x18,x19,x20,x21,x22,x23,x24,
    x25,x26,x27,x28,x29,x30,x31,x32 be Element of S;

    consider a1 be FinSequence of S such that
    A1:a1 is 16-element &
    a1.1=x1&a1.2=x2&a1.3=x3&a1.4=x4&
    a1.5=x5&a1.6=x6&a1.7=x7&a1.8=x8&
    a1.9=x9&a1.10=x10&a1.11=x11&a1.12=x12&
    a1.13=x13&a1.14=x14&a1.15=x15&a1.16=x16 by Th21;

    consider a2 be FinSequence of S such that
    A2:a2 is 16-element &
    a2.1=x17&a2.2=x18&a2.3=x19&a2.4=x20&
    a2.5=x21&a2.6=x22&a2.7=x23&a2.8=x24&
    a2.9=x25&a2.10=x26&a2.11=x27&a2.12=x28&
    a2.13=x29&a2.14=x30&a2.15=x31&a2.16=x32 by Th21;
     reconsider a1,a2 as 16-element FinSequence of S by A1,A2;
    take a1^a2;
    thus a1^a2 is 32-element;
A3:   (a1^a2).1 = a1.1 & ... & (a1^a2).16 = a1.16 by FINSEQ_3:154;
     (a1^a2).(16+1) = a2.1 & ... & (a1^a2).(16+16) = a2.16
       by FINSEQ_3:155;
   hence thesis by A3,A1,A2;
  end;
