reserve
  a for natural Number,
  k,l,m,n,k1,b,c,i for Nat,
  x,y,z,y1,y2 for object,
  X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for FinSequence;
reserve D for set;

theorem Th70:
  for x1,x2,x3,x4, x5, x6, x7, x8 being set, p being FinSequence
  st p = <*x1*>^<*x2*>^<*x3*>^<*x4*>^<*x5*>^<*x6*>^<*x7*>^<*x8*>
  holds len p = 8 & p.1 = x1 & p.2 = x2 & p.3 = x3 & p.4 = x4 & p.5 = x5 &
  p.6 = x6 & p.7 = x7 & p.8 = x8
proof
  let x1, x2, x3, x4, x5, x6, x7, x8 be set, p be FinSequence;
  thus p = <*x1*>^<*x2*>^<*x3*>^<*x4*>^<*x5*>^<*x6*>^<*x7*>^<*x8*>
  implies len p = 8 & p.1 = x1 & p.2 = x2 & p.3 = x3 & p.4 = x4 & p.5 = x5 &
  p.6 = x6 & p.7 = x7 & p.8 = x8
  proof
  assume
A1: p = <*x1*>^<*x2*>^<*x3*>^<*x4*>^<*x5*>^<*x6*>^<*x7*>^<*x8*>;
  set p17 = <*x1*>^<*x2*>^<*x3*>^<*x4*>^<*x5*>^<*x6*>^<*x7*>;
A2: len p17 = 7 by Th69;
A3: p17.1 = x1 & p17.2 = x2 by Th69;
A4: p17.3 = x3 & p17.4 = x4 by Th69;
A5: p17.5 = x5 & p17.6 = x6 by Th69;
A6: p17.7 = x7 by Th69;
  thus len p = len p17 + len <*x8*> by A1,Th22
    .= 7 + 1 by A2,Th40
    .= 8;
A7:  dom p17 = Seg 7 by A2,Def3;
   1 in Seg 7 & ...& 7 in Seg 7;
  hence p.1 = x1 & p.2 = x2 & p.3 = x3 & p.4 = x4 & p.5 = x5 & p.6 = x6 &
  p.7 = x7 by A1,A3,A4,A5,A6,Def7,A7;
  thus p.8 = p.(len p17 + 1) by A2
    .= x8 by A1,Th42;
  end;
end;
