let IAlph, OAlph be non empty set ; :: thesis: for tfsm being non empty Mealy-FSM of IAlph,OAlph
for qa, qb being State of tfsm
for k being Nat st k -equivalent qa,qb holds
k -equivalent qb,qa
let tfsm be non empty Mealy-FSM of IAlph,OAlph; :: thesis: for qa, qb being State of tfsm
for k being Nat st k -equivalent qa,qb holds
k -equivalent qb,qa
let qa, qb be State of tfsm; :: thesis: for k being Nat st k -equivalent qa,qb holds
k -equivalent qb,qa
let k be Nat; :: thesis: ( k -equivalent qa,qb implies k -equivalent qb,qa )
assume A1: k -equivalent qa,qb ; :: thesis: k -equivalent qb,qa
thus k -equivalent qb,qa :: thesis: verum

proof

let w be FinSequence of IAlph; :: according to FSM_1:def 11 :: thesis: ( len w <= k implies qb,w -response = qa,w -response )
assume len w <= k ; :: thesis: qb,w -response = qa,w -response
hence qb,w -response = qa,w -response by A1, Def11; :: thesis: verum

end;