let k be Nat; :: thesis: for N being with_non-empty_elements set
for S being non empty IC-Ins-separated Mem-Struct of N
for s being State of S
for p being PartState of S st p +* (Start-At (k,S)) c= s holds
IC s = k
let N be with_non-empty_elements set ; :: thesis: for S being non empty IC-Ins-separated Mem-Struct of N
for s being State of S
for p being PartState of S st p +* (Start-At (k,S)) c= s holds
IC s = k
let S be non empty IC-Ins-separated Mem-Struct of N; :: thesis: for s being State of S
for p being PartState of S st p +* (Start-At (k,S)) c= s holds
IC s = k
let s be State of S; :: thesis: for p being PartState of S st p +* (Start-At (k,S)) c= s holds
IC s = k
let p be PartState of S; :: thesis: ( p +* (Start-At (k,S)) c= s implies IC s = k )
assume A1: p +* (Start-At (k,S)) c= s ; :: thesis: IC s = k
IC in dom (p +* (Start-At (k,S))) by Th141;
hence IC s = IC (p +* (Start-At (k,S))) by A1, GRFUNC_1:2
.= k by Lm142 ;
:: thesis: verum