reserve x,N for set,
        k for Nat;
reserve N for with_zero set;
reserve S for IC-Ins-separated non empty with_non-empty_values
     Mem-Struct over N;
reserve s for State of S;
reserve p for PartState of S;

theorem
  for S being IC-Ins-separated non empty with_non-empty_values
               Mem-Struct over N,
      l being Nat,
      p being l-started PartState of S
  for s being PartState of S st p c= s holds s is l-started
proof
  let S be IC-Ins-separated non empty with_non-empty_values Mem-Struct over N,
      l be Nat,
      p be l-started PartState of S;
A1: IC S in dom p by Def11;
A2: IC p = l by Def11;
  let s be PartState of S;
  assume
A3:  p c= s;
   then dom p c= dom s by RELAT_1:11;
  hence IC S in dom s by A1;
  thus IC s = l by A3,A2,A1,GRFUNC_1:2;
end;
