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 d being data-only PartState of S, k being Nat
  holds IncIC(p+*d,k) = IncIC(p,k) +* d
proof
  let d be data-only PartState of S, k be Nat;
A1: d tolerates Start-At(IC p+k,S) by Th23;
  thus IncIC(p+*d,k) = p +* d +* Start-At(IC p+k,S) by Th11
    .= p +* (d +* Start-At(IC p+k,S)) by FUNCT_4:14
    .= p +* (Start-At(IC p+k,S) +* d) by A1,FUNCT_4:34
    .= IncIC(p,k) +* d by FUNCT_4:14;
end;
