let n be non empty Element of NAT ; :: thesis: for m, k1, k2 being FinSequence of NAT st (2 to_power n) + 1 is prime & len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n & k1 . 1 is_expressible_by n & k1 . 2 is_expressible_by n & k1 . 3 is_expressible_by n & k1 . 4 is_expressible_by n & k2 . 1 = INV_MOD (k1 . 1),n & k2 . 2 = NEG_MOD (k1 . 2),n & k2 . 3 = NEG_MOD (k1 . 3),n & k2 . 4 = INV_MOD (k1 . 4),n holds
((IDEA_QS k2,n) * (IDEA_PS k1,n)) . m = m
let m, k1, k2 be FinSequence of NAT ; :: thesis: ( (2 to_power n) + 1 is prime & len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n & k1 . 1 is_expressible_by n & k1 . 2 is_expressible_by n & k1 . 3 is_expressible_by n & k1 . 4 is_expressible_by n & k2 . 1 = INV_MOD (k1 . 1),n & k2 . 2 = NEG_MOD (k1 . 2),n & k2 . 3 = NEG_MOD (k1 . 3),n & k2 . 4 = INV_MOD (k1 . 4),n implies ((IDEA_QS k2,n) * (IDEA_PS k1,n)) . m = m )
assume that
A1: (2 to_power n) + 1 is prime and
A2: len m >= 4 and
A3: ( m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n ) and
A4: ( k1 . 1 is_expressible_by n & k1 . 2 is_expressible_by n & k1 . 3 is_expressible_by n & k1 . 4 is_expressible_by n ) and
A5: ( k2 . 1 = INV_MOD (k1 . 1),n & k2 . 2 = NEG_MOD (k1 . 2),n & k2 . 3 = NEG_MOD (k1 . 3),n & k2 . 4 = INV_MOD (k1 . 4),n ) ; :: thesis: ((IDEA_QS k2,n) * (IDEA_PS k1,n)) . m = m
( dom (IDEA_PS k1,n) = MESSAGES & dom (IDEA_QS k2,n) = MESSAGES ) by FUNCT_2:def 1;
then ( m in dom (IDEA_PS k1,n) & m in dom (IDEA_QS k2,n) ) by FINSEQ_1:def 11;
then ((IDEA_QS k2,n) * (IDEA_PS k1,n)) . m = (IDEA_QS k2,n) . ((IDEA_PS k1,n) . m) by FUNCT_1:23
.= (IDEA_QS k2,n) . (IDEAoperationA m,k1,n) by Def19
.= IDEAoperationA (IDEAoperationA m,k1,n),k2,n by Def20
.= m by A1, A2, A3, A4, A5, Th30 ;
hence ((IDEA_QS k2,n) * (IDEA_PS k1,n)) . m = m ; :: thesis: verum