let n be non zero 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) & k2 . 5 = k1 . 5 & k2 . 6 = k1 . 6 holds
((IDEA_QE (k2,n)) * (IDEA_PE (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) & k2 . 5 = k1 . 5 & k2 . 6 = k1 . 6 implies ((IDEA_QE (k2,n)) * (IDEA_PE (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 & 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) ) and
A5: ( k2 . 5 = k1 . 5 & k2 . 6 = k1 . 6 ) ; :: thesis: ((IDEA_QE (k2,n)) * (IDEA_PE (k1,n))) . m = m
A6: ( (IDEAoperationB (m,k1,n)) . 2 is_expressible_by n & (IDEAoperationB (m,k1,n)) . 3 is_expressible_by n ) by A2, Th27;
A7: (IDEAoperationB (m,k1,n)) . 4 is_expressible_by n by A2, Th27;
A8: ( len (IDEAoperationB (m,k1,n)) >= 4 & (IDEAoperationB (m,k1,n)) . 1 is_expressible_by n ) by A2, Def12, Th27;
dom (IDEA_PE (k1,n)) = MESSAGES by FUNCT_2:def 1;
then m in dom (IDEA_PE (k1,n)) by FINSEQ_1:def 11;
then ((IDEA_QE (k2,n)) * (IDEA_PE (k1,n))) . m = (IDEA_QE (k2,n)) . ((IDEA_PE (k1,n)) . m) by FUNCT_1:13
.= (IDEA_QE (k2,n)) . (IDEAoperationA ((IDEAoperationB (m,k1,n)),k1,n)) by Def21
.= IDEAoperationB ((IDEAoperationA ((IDEAoperationA ((IDEAoperationB (m,k1,n)),k1,n)),k2,n)),k2,n) by Def22
.= IDEAoperationB ((IDEAoperationB (m,k1,n)),k2,n) by A1, A4, A8, A6, A7, Th29
.= m by A2, A3, A5, Th31 ;
hence ((IDEA_QE (k2,n)) * (IDEA_PE (k1,n))) . m = m ; :: thesis: verum