let I be non empty set ; :: thesis: for s1, s2 being Element of I
for S being non empty FSM of I
for q being State of S holds GEN <*s1,s2*>,q = <*q,(the Tran of S . [q,s1]),(the Tran of S . [(the Tran of S . [q,s1]),s2])*>
let s1, s2 be Element of I; :: thesis: for S being non empty FSM of I
for q being State of S holds GEN <*s1,s2*>,q = <*q,(the Tran of S . [q,s1]),(the Tran of S . [(the Tran of S . [q,s1]),s2])*>
let S be non empty FSM of I; :: thesis: for q being State of S holds GEN <*s1,s2*>,q = <*q,(the Tran of S . [q,s1]),(the Tran of S . [(the Tran of S . [q,s1]),s2])*>
let q be State of S; :: thesis: GEN <*s1,s2*>,q = <*q,(the Tran of S . [q,s1]),(the Tran of S . [(the Tran of S . [q,s1]),s2])*>
set w = <*s1,s2*>;
A1: (GEN <*s1,s2*>,q) . 1 = q by FSM_1:def 2;
A2: <*s1,s2*> = <*s1*> ^ <*s2*> by FINSEQ_1:def 9;
A3: len <*s1*> = 1 by FINSEQ_1:56;
GEN <*s1*>,q = <*q,(the Tran of S . [q,s1])*> by Th1;
then (GEN <*s1*>,q) . (1 + 1) = the Tran of S . [q,s1] by FINSEQ_1:61;
then q,<*s1*> -leads_to the Tran of S . [q,s1] by A3, FSM_1:def 3;
then A4: (GEN <*s1,s2*>,q) . (1 + 1) = the Tran of S . [q,s1] by A2, A3, FSM_1:21;
A5: len <*s1,s2*> = 2 by FINSEQ_1:61;
2 <= len <*s1,s2*> by FINSEQ_1:61;
then consider WI being Element of I, QI, QI1 being State of S such that
A6: WI = <*s1,s2*> . 2 and
A7: QI = (GEN <*s1,s2*>,q) . 2 and
A8: QI1 = (GEN <*s1,s2*>,q) . (2 + 1) and
A9: WI -succ_of QI = QI1 by FSM_1:def 2;
A10: WI = s2 by A6, FINSEQ_1:61;
len (GEN <*s1,s2*>,q) = (len <*s1,s2*>) + 1 by FSM_1:def 2
.= 3 by A5 ;
hence GEN <*s1,s2*>,q = <*q,(the Tran of S . [q,s1]),(the Tran of S . [(the Tran of S . [q,s1]),s2])*> by A1, A4, A7, A8, A9, A10, FINSEQ_1:62; :: thesis: verum