let x, y be set ; :: thesis: for E being non empty set
for F being Subset of (E ^omega )
for TS being transition-system of F st not <%> E in rng (dom the Tran of TS) holds
not x, <%> E -->. y,TS
let E be non empty set ; :: thesis: for F being Subset of (E ^omega )
for TS being transition-system of F st not <%> E in rng (dom the Tran of TS) holds
not x, <%> E -->. y,TS
let F be Subset of (E ^omega ); :: thesis: for TS being transition-system of F st not <%> E in rng (dom the Tran of TS) holds
not x, <%> E -->. y,TS
let TS be transition-system of F; :: thesis: ( not <%> E in rng (dom the Tran of TS) implies not x, <%> E -->. y,TS )
assume A: not <%> E in rng (dom the Tran of TS) ; :: thesis: not x, <%> E -->. y,TS
assume x, <%> E -->. y,TS ; :: thesis: contradiction
then [[x,(<%> E)],y] in the Tran of TS by DefProd;
then [x,(<%> E)] in dom the Tran of TS by RELAT_1:20;
hence contradiction by A, RELAT_1:20; :: thesis: verum