reserve i,j,k,n for Nat;
reserve x,y,z for Tuple of n, BOOLEAN;
reserve m,k,k1,k2 for FinSequence of NAT;

theorem Th27:
  for n being non zero Nat holds len m >= 4 implies
  IDEAoperationB(m,k,n).1 is_expressible_by n & IDEAoperationB(m,k,n).2
  is_expressible_by n & IDEAoperationB(m,k,n).3 is_expressible_by n &
  IDEAoperationB(m,k,n).4 is_expressible_by n
proof
  let n be non zero Nat;
  assume
A1: len m >= 4;
  then 1 <= len m by XXREAL_0:2;
  then 1 in Seg len m by FINSEQ_1:1;
  then 1 in dom m by FINSEQ_1:def 3;
  then IDEAoperationB(m,k,n).1 = Absval((n-BinarySequence m.1) 'xor' (n
-BinarySequence MUL_MOD(ADD_MOD(MUL_MOD(Absval((n-BinarySequence m.1) 'xor' (n
  -BinarySequence m.3)),k.5,n),Absval((n-BinarySequence m.2) 'xor' (n
  -BinarySequence m.4)),n),k.6,n))) by Def12;
  then
A2: IDEAoperationB(m,k,n).1 < 2 to_power n by BINARI_3:1;
  3 <= len m by A1,XXREAL_0:2;
  then 3 in Seg len m by FINSEQ_1:1;
  then 3 in dom m by FINSEQ_1:def 3;
  then IDEAoperationB(m,k,n).3 = Absval((n-BinarySequence m.3) 'xor' (n
-BinarySequence MUL_MOD(ADD_MOD(MUL_MOD(Absval((n-BinarySequence m.1) 'xor' (n
  -BinarySequence m.3)),k.5,n),Absval((n-BinarySequence m.2) 'xor' (n
  -BinarySequence m.4)),n),k.6,n))) by Def12;
  then
A3: IDEAoperationB(m,k,n).3 < 2 to_power n by BINARI_3:1;
  2 <= len m by A1,XXREAL_0:2;
  then 2 in Seg len m by FINSEQ_1:1;
  then 2 in dom m by FINSEQ_1:def 3;
  then IDEAoperationB(m,k,n).2 = Absval((n-BinarySequence m.2) 'xor' (n
  -BinarySequence ADD_MOD(MUL_MOD(Absval((n-BinarySequence m.1) 'xor' (n
-BinarySequence m.3)),k.5,n),MUL_MOD(ADD_MOD(MUL_MOD( Absval((n-BinarySequence
m.1) 'xor' (n-BinarySequence m.3)), k.5,n),Absval((n-BinarySequence m.2) 'xor'
  (n-BinarySequence m.4)),n),k.6,n),n))) by Def12;
  then
A4: IDEAoperationB(m,k,n).2 < 2 to_power n by BINARI_3:1;
  4 in Seg len m by A1,FINSEQ_1:1;
  then 4 in dom m by FINSEQ_1:def 3;
  then IDEAoperationB(m,k,n).4 = Absval((n-BinarySequence m.4) 'xor' (n
  -BinarySequence ADD_MOD(MUL_MOD(Absval((n-BinarySequence m.1) 'xor' (n
-BinarySequence m.3)),k.5,n),MUL_MOD(ADD_MOD(MUL_MOD( Absval((n-BinarySequence
m.1) 'xor' (n-BinarySequence m.3)), k.5,n),Absval((n-BinarySequence m.2) 'xor'
  (n-BinarySequence m.4)),n),k.6,n),n))) by Def12;
  then IDEAoperationB(m,k,n).4 < 2 to_power n by BINARI_3:1;
  hence thesis by A2,A4,A3;
end;
