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 Th45:
  for x1, x2, x3, x4, x5, x6, x7 being set st
     p = <%x1%>^<%x2%>^<%x3%>^<%x4%>^<%x5%>^<%x6%>^<%x7%>
  holds len p = 7 & p.0 = x1 & p.1 = x2 & p.2 = x3 & p.3 = x4 & p.4 = x5 &
  p.5 = x6 & p.6 = x7
proof
  let x1, x2, x3, x4, x5, x6, x7 be set;
  assume
A1: p = <%x1%>^<%x2%>^<%x3%>^<%x4%>^<%x5%>^<%x6%>^<%x7%>;
  set p16 = <%x1%>^<%x2%>^<%x3%>^<%x4%>^<%x5%>^<%x6%>;
A2: len p16 = 6 by Th44;
A3: p16.0 = x1 & p16.1 = x2 by Th44;
A4: p16.2 = x3 & p16.3 = x4 by Th44;
A5: p16.4 = x5 & p16.5 = x6 by Th44;
  thus len p = len p16 + len <%x7%> by A1,Def3
    .= 6 + 1 by A2,Th30
    .= 7;
   0 in 6 & ... & 5 in 6 by CARD_1:54,ENUMSET1:def 4;
  hence p.0 = x1 & p.1 = x2 & p.2 = x3 & p.3 = x4 & p.4 = x5 & p.5 = x6
  by A1,A3,A4,A5,Def3,A2;
  thus p.6 = p.len p16 by Th44
    .= x7 by A1,Th33;
end;
