reserve i,n for Nat;
reserve m for non zero Nat;
reserve p,q for Tuple of n, BOOLEAN;
reserve d,d1,d2 for Element of BOOLEAN;

theorem Th10:
  for z being Tuple of m, BOOLEAN for d being Element of BOOLEAN holds
  Intval(z^<*d*>) =
  Absval(z)-(IFEQ(d,FALSE,0,2 to_power(m)) qua Nat)
proof
  let z be Tuple of m, BOOLEAN;
  let d;
  per cases by XBOOLEAN:def 3;
  suppose
A1: d = FALSE;
then  (z^<*d*>)/.(m+1) = FALSE by BINARITH:2;
then A2: Intval(z^<*d*>) = Absval(z^<*d*>) by Def3
      .= Absval(z)+IFEQ(d,FALSE,0,2 to_power(m)) by BINARITH:20
      .= Absval(z)+0 by A1,FUNCOP_1:def 8
      .= Absval(z);
 Absval(z)-(IFEQ(d,FALSE,0,2 to_power(m)) qua Nat)
    = Absval(z) - 0 by A1,FUNCOP_1:def 8
      .= Absval(z);
    hence thesis by A2;
  end;
  suppose
A3: d = TRUE;
then  (z^<*d*>)/.(m+1) <> FALSE by BINARITH:2;
then  Intval(z^<*d*>) = Absval(z^<*d*>) - 2 to_power(m+1) by Def3
      .= Absval(z) + IFEQ(d,FALSE,0,2 to_power m)-2 to_power(m+1)
    by BINARITH:20
      .= Absval(z) + 2 to_power m - 2 to_power(m+1)
    by A3,FUNCOP_1:def 8
      .= Absval(z) + 2 to_power m - (2 to_power m * 2 to_power 1) by POWER:27
      .= Absval(z) + 2 to_power m - 2*2 to_power m by POWER:25
      .= Absval(z) - 2 to_power m;
    hence thesis by A3,FUNCOP_1:def 8;
  end;
end;
