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 Th47:
 for p being PartState of S
 holds Initialize p c= s implies IC s = 0
proof let p be PartState of S;
A1: IC Initialize p =  0 by Def11;
A2: IC S in dom Initialize p by Def11;
  assume Initialize p c= s;
  hence thesis by A1,A2,GRFUNC_1:2;
end;
