reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;

theorem Th46:
  for x1,x2,x3,x4, x5, x6, x7, x8 being set st
    p = <%x1%>^<%x2%>^<%x3%>^<%x4%>^<%x5%>^<%x6%>^<%x7%>^<%x8%>
  holds len p = 8 & p.0 = x1 & p.1 = x2 & p.2 = x3 & p.3 = x4 & p.4 = x5 &
  p.5 = x6 & p.6 = x7 & p.7 = x8
proof
  let x1, x2, x3, x4, x5, x6, x7, x8 be set;
  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 Th45;
A3: p17.0 = x1 & p17.1 = x2 by Th45;
A4: p17.2 = x3 & p17.3 = x4 by Th45;
A5: p17.4 = x5 & p17.5 = x6 by Th45;
A6: p17.6 = x7 by Th45;
  thus len p = len p17 + len <%x8%> by A1,Def3
    .= 7 + 1 by A2,Th30
    .= 8;
   0 in 7 & ... & 6 in 7 by CARD_1:55,ENUMSET1:def 5;
  hence p.0 = x1 & p.1 = x2 & p.2 = x3 & p.3 = x4 & p.4 = x5 & p.5 = x6 &
  p.6 = x7 by A1,A3,A4,A5,A6,Def3,A2;
  thus p.7 = p.len p17 by Th45
    .= x8 by A1,Th33;
end;
