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 Th26:
 for p being PartState of S st IC S in dom p
  holds p = DataPart p +* Start-At (IC p,S)
proof
 let p be PartState of S;
A2: dom DataPart p misses dom Start-At (IC p,S) by Th3,ZFMISC_1:50;
A3: dom((Start-At(IC p,S))) misses dom DataPart p
                  by Th3,ZFMISC_1:50;
 assume IC S in dom p;
  then p = Start-At(IC p,S) +* DataPart p by Th19;
  then
A4:  p = (Start-At(IC p,S) ) \/  DataPart p by A3,FUNCT_4:31;
 thus thesis by A2,A4,FUNCT_4:31;
end;
