let x, y be set ; :: thesis: for E being non empty set
for u, v being Element of E ^omega
for F being Subset of (E ^omega )
for TS being transition-system of F st not <%> E in rng (dom the Tran of TS) & x,u ==>. y,v,TS holds
len u > len v
let E be non empty set ; :: thesis: for u, v being Element of E ^omega
for F being Subset of (E ^omega )
for TS being transition-system of F st not <%> E in rng (dom the Tran of TS) & x,u ==>. y,v,TS holds
len u > len v
let u, v be Element of E ^omega ; :: 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) & x,u ==>. y,v,TS holds
len u > len v
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) & x,u ==>. y,v,TS holds
len u > len v
let TS be transition-system of F; :: thesis: ( not <%> E in rng (dom the Tran of TS) & x,u ==>. y,v,TS implies len u > len v )
assume A: not <%> E in rng (dom the Tran of TS) ; :: thesis: ( not x,u ==>. y,v,TS or len u > len v )
assume B: x,u ==>. y,v,TS ; :: thesis: len u > len v
consider w being Element of E ^omega such that
C: ( x,w -->. y,TS & u = w ^ v ) by B, ThDir25;
D: w in rng (dom the Tran of TS) by C, ThProd30;

per cases ( v = <%> E or v <> <%> E ) ;

suppose E: v = <%> E ; :: thesis: len u > len v

then u <> <%> E by A, B, ThDir60;
then len u > 0 ;
hence len u > len v by E; :: thesis: verum

end;

suppose v <> <%> E ; :: thesis: len u > len v

hence len u > len v by A, C, D, LmXSeq60; :: thesis: verum

end;

end;