let x, y be set ; :: thesis: for E being non empty set
for e being Element of E
for F being Subset of (E ^omega )
for TS being non empty transition-system of F st not <%> E in rng (dom the Tran of TS) & ==>.-relation TS reduces [x,<%e%>],[y,(<%> E)] holds
[[x,<%e%>],[y,(<%> E)]] in ==>.-relation TS
let E be non empty set ; :: thesis: for e being Element of E
for F being Subset of (E ^omega )
for TS being non empty transition-system of F st not <%> E in rng (dom the Tran of TS) & ==>.-relation TS reduces [x,<%e%>],[y,(<%> E)] holds
[[x,<%e%>],[y,(<%> E)]] in ==>.-relation TS
let e be Element of E; :: thesis: for F being Subset of (E ^omega )
for TS being non empty transition-system of F st not <%> E in rng (dom the Tran of TS) & ==>.-relation TS reduces [x,<%e%>],[y,(<%> E)] holds
[[x,<%e%>],[y,(<%> E)]] in ==>.-relation TS
let F be Subset of (E ^omega ); :: thesis: for TS being non empty transition-system of F st not <%> E in rng (dom the Tran of TS) & ==>.-relation TS reduces [x,<%e%>],[y,(<%> E)] holds
[[x,<%e%>],[y,(<%> E)]] in ==>.-relation TS
let TS be non empty transition-system of F; :: thesis: ( not <%> E in rng (dom the Tran of TS) & ==>.-relation TS reduces [x,<%e%>],[y,(<%> E)] implies [[x,<%e%>],[y,(<%> E)]] in ==>.-relation TS )
assume A: not <%> E in rng (dom the Tran of TS) ; :: thesis: ( not ==>.-relation TS reduces [x,<%e%>],[y,(<%> E)] or [[x,<%e%>],[y,(<%> E)]] in ==>.-relation TS )
assume ==>.-relation TS reduces [x,<%e%>],[y,(<%> E)] ; :: thesis: [[x,<%e%>],[y,(<%> E)]] in ==>.-relation TS
then consider P being RedSequence of ==>.-relation TS such that
B: ( P . 1 = [x,<%e%>] & P . (len P) = [y,(<%> E)] ) by REWRITE1:def 3;
C: len P = 1 + 1 by A, B, ThRedSeq100;
then ( 1 in dom P & 1 + 1 in dom P ) by FINSEQ_3:27;
hence [[x,<%e%>],[y,(<%> E)]] in ==>.-relation TS by B, C, REWRITE1:def 2; :: thesis: verum