let x, y be object ; 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 over F st not <%> E in rng (dom the Tran of TS) & x,<%e%> ==>* y, <%> E,TS holds
x,<%e%> ==>. y, <%> E,TS
let E be non empty set ; for e being Element of E
for F being Subset of (E ^omega)
for TS being non empty transition-system over F st not <%> E in rng (dom the Tran of TS) & x,<%e%> ==>* y, <%> E,TS holds
x,<%e%> ==>. y, <%> E,TS
let e be Element of E; for F being Subset of (E ^omega)
for TS being non empty transition-system over F st not <%> E in rng (dom the Tran of TS) & x,<%e%> ==>* y, <%> E,TS holds
x,<%e%> ==>. y, <%> E,TS
let F be Subset of (E ^omega); for TS being non empty transition-system over F st not <%> E in rng (dom the Tran of TS) & x,<%e%> ==>* y, <%> E,TS holds
x,<%e%> ==>. y, <%> E,TS
let TS be non empty transition-system over F; ( not <%> E in rng (dom the Tran of TS) & x,<%e%> ==>* y, <%> E,TS implies x,<%e%> ==>. y, <%> E,TS )
assume A1:
not <%> E in rng (dom the Tran of TS)
; ( not x,<%e%> ==>* y, <%> E,TS or x,<%e%> ==>. y, <%> E,TS )
assume
x,<%e%> ==>* y, <%> E,TS
; x,<%e%> ==>. y, <%> E,TS
then
==>.-relation TS reduces [x,<%e%>],[y,(<%> E)]
;
then
[[x,<%e%>],[y,(<%> E)]] in ==>.-relation TS
by A1, Th79;
hence
x,<%e%> ==>. y, <%> E,TS
by Def4; verum