reserve n for non zero Nat,
  j,k,l,m for Nat,
  g,h,i for Integer;

theorem
  for n be non zero Nat st -2 to_power ((n+1)-'1) <= h + i & h < 0 & i
  < 0 & -2 to_power (n-'1) <= h & -2 to_power (n-'1) <= i holds Intval(
  2sComplement(n+1,h) + 2sComplement(n+1,i)) = h+i
proof
  let n be non zero Nat such that
A1: -2 to_power ((n+1)-'1) <= h + i and
A2: h < 0 and
A3: i < 0 and
A4: -2 to_power (n-'1) <= h & -2 to_power (n-'1) <= i;
A5: 2 to_power (n+1) + -(2 to_power n) = -(2 to_power n) + ((2 to_power 1)*
  (2 to_power n)) by POWER:27
    .= -(2 to_power n) + 2*(2 to_power n) by POWER:25
    .= 2 to_power n;
  (n+1)-1 = n;
  then
A6: -2 to_power n <= h+i by A1,XREAL_0:def 2;
  then
A7: -2 to_power n < h by A2,A3,Th9;
  then
A8: 2 to_power n <= 2 to_power (n+1) + h by A5,XREAL_1:6;
A9: -2 to_power n < i by A2,A3,A6,Th9;
  then
A10: 0 <= 2 to_power (n+1) + i by A5,XREAL_1:6;
  0 <= 2 to_power (n+1) + h by A7,A5,XREAL_1:6;
  then reconsider
  NH = 2 to_power (n+1) + h, NI = 2 to_power (n+1) + i as Element
  of NAT by A10,INT_1:3;
A11: 1 <= n + 1 by NAT_1:11;
  set H = 2sComplement(n,h), I = 2sComplement(n,i), H1 = 2sComplement(n+1,h),
  I1 = 2sComplement(n+1,i), F = FALSE, T = TRUE;
  n < n+1 by XREAL_1:29;
  then
A12: 2 to_power n < 2 to_power (n+1) by POWER:39;
A13: (ex a be Element of BOOLEAN st H1 = H^<*a*> )& ex a be Element of
  BOOLEAN st I1 = I^<*a*> by Th33;
A14: 2 to_power (n+1) + h < 2 to_power (n+1) + 0 by A2,XREAL_1:8;
  -h < --2 to_power n by A7,XREAL_1:24;
  then |.h.| < 2 to_power n by A2,ABSVALUE:def 1;
  then
A15: MajP(n+1,|.h.|) = n+1 by A12,Th24,XXREAL_0:2;
  then
A16: H1 = (n+1)-BinarySequence(|.2 to_power (n+1)+h.|) by A2,Def2
    .= (n+1)-BinarySequence(NH) by ABSVALUE:def 1;
  len H1 = n + 1 by CARD_1:def 7;
  then
A17: H1/.(n+1) = H1.(n+1) by A11,FINSEQ_4:15
    .= ((n+1)-BinarySequence(|.2 to_power (n+1)+h.|)).(n+1) by A2,A15,Def2
    .= ((n+1)-BinarySequence(NH)).(n+1) by ABSVALUE:def 1
    .= T by A14,A8,BINARI_3:29;
A18: 2 to_power n <= 2 to_power (n+1) + i by A9,A5,XREAL_1:6;
A19: 2 to_power (n+1) + i < 2 to_power (n+1) + 0 by A3,XREAL_1:8;
  -i < --2 to_power n by A9,XREAL_1:24;
  then |.i.| < 2 to_power n by A3,ABSVALUE:def 1;
  then
A20: MajP(n+1,|.i.|) = n+1 by A12,Th24,XXREAL_0:2;
  then
A21: I1 = (n+1)-BinarySequence(|.2 to_power (n+1)+i.|) by A3,Def2
    .= (n+1)-BinarySequence(NI) by ABSVALUE:def 1;
  len I1 = n + 1 by CARD_1:def 7;
  then
A22: I1/.(n+1) = I1.(n+1) by A11,FINSEQ_4:15
    .= ((n+1)-BinarySequence(|.2 to_power (n+1)+i.|)).(n+1) by A3,A20,Def2
    .= ((n+1)-BinarySequence(NI)).(n+1) by ABSVALUE:def 1
    .= T by A19,A18,BINARI_3:29;
  then
A23: Intval(I1) = Absval(I1) - 2 to_power (n+1) by BINARI_2:def 3
    .= NI - 2 to_power (n+1) by A19,A21,BINARI_3:35
    .= i;
A24: carry(H1,I1)/.(n+1) = T by A2,A3,A4,A6,Th35;
  then
A25: Int_add_ovfl(H1,I1) = F '&' 'not' T '&' T by A17,A22,BINARI_2:def 4
    .= F;
A26: Int_add_udfl(H1,I1) = T '&' T '&' 'not' T by A17,A22,A24,BINARI_2:def 5
    .= F;
  Intval(H1) = Absval(H1) - 2 to_power (n+1) by A17,BINARI_2:def 3
    .= NH - 2 to_power (n+1) by A14,A16,BINARI_3:35
    .= h;
  then Intval(H1+I1) = h + i - IFEQ(F,F,0,2 to_power(n+1)) + IFEQ(F,F,0,2
  to_power(n+1)) by A13,A23,A25,A26,BINARI_2:12
    .= h + i - 0 + 0;
  hence thesis;
end;
