let x, y be object ; :: 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 over 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 over 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 over 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 over 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 over F; :: thesis: ( not <%> E in rng (dom the Tran of TS) & x,u ==>. y,v,TS implies len u > len v )
assume A1: not <%> E in rng (dom the Tran of TS) ; :: thesis: ( not x,u ==>. y,v,TS or len u > len v )
assume A2: x,u ==>. y,v,TS ; :: thesis: len u > len v
then consider w being Element of E ^omega such that
A3: x,w -->. y,TS and
A4: u = w ^ v ;
A5: w in rng (dom the Tran of TS) by A3, Th15;

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

suppose A6: v = <%> E ; :: thesis: len u > len v

then u <> <%> E by A1, A2, Th28;
then len u > 0 ;
hence len u > len v by A6; :: thesis: verum

end;

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

hence len u > len v by A1, A4, A5, Th10; :: thesis: verum

end;

end;