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 Th43:
  for x1, x2, x3, x4, x5 being set st
     p = <%x1%>^<%x2%>^<%x3%>^<%x4%>^<%x5%>
  holds len p = 5 & p.0 = x1 & p.1 = x2 & p.2 = x3 & p.3 = x4 & p.4 = x5
proof
  let x1, x2, x3, x4, x5 be set;
  assume
A1: p = <%x1%>^<%x2%>^<%x3%>^<%x4%>^<%x5%>;
  set p14 = <%x1%>^<%x2%>^<%x3%>^<%x4%>;
A2: len p14 = 4 by Th42;
A3: p14.0 = x1 & p14.1 = x2 by Th42;
A4: p14.2 = x3 & p14.3 = x4 by Th42;
  thus len p = len p14 + len <%x5%> by A1,Def3
    .= 4 + 1 by A2,Th30
    .= 5;
   0 in 4 & ... & 3 in 4 by CARD_1:52,ENUMSET1:def 2;
  hence p.0 = x1 & p.1 = x2 & p.2 = x3 & p.3 = x4 by A1,A3,A4,Def3,A2;
  thus p.4 = p.len p14 by Th42
    .= x5 by A1,Th33;
end;
