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) & x,<%e%> ==>* y, <%> E,TS holds
x,<%e%> ==>. y, <%> E,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) & x,<%e%> ==>* y, <%> E,TS holds
x,<%e%> ==>. y, <%> E,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) & x,<%e%> ==>* y, <%> E,TS holds
x,<%e%> ==>. y, <%> E,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) & x,<%e%> ==>* y, <%> E,TS holds
x,<%e%> ==>. y, <%> E,TS
let TS be non empty transition-system of F; :: thesis: ( 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) ; :: thesis: ( not x,<%e%> ==>* y, <%> E,TS or x,<%e%> ==>. y, <%> E,TS )
assume x,<%e%> ==>* y, <%> E,TS ; :: thesis: x,<%e%> ==>. y, <%> E,TS
then ==>.-relation TS reduces [x,<%e%>],[y,(<%> E)] by DefTran;
then [[x,<%e%>],[y,(<%> E)]] in ==>.-relation TS by A1, ThRed130;
hence x,<%e%> ==>. y, <%> E,TS by DefRel; :: thesis: verum