
theorem Th18:
  for G being real-weighted WGraph, src being Vertex of G, i,j
  being Nat st i <= j holds dom ((DIJK:CompSeq(src).i))`1 c= dom (DIJK:CompSeq(
  src).j)`1 & (DIJK:CompSeq(src).i)`2 c= (DIJK:CompSeq(src).j)`2
proof
  let G be real-weighted WGraph, src be Vertex of G, i,j be Nat;
  set DCS = DIJK:CompSeq(src);
  set dDCS = dom (DCS.i)`1;
  set eDCS = (DCS.i)`2;
  defpred P[Nat] means dDCS c= dom (DCS.(i+$1))`1 & eDCS c= (DCS.(i+$1))`2;
  assume i <= j;
  then
A1: ex x being Nat st j = i + x by NAT_1:10;
  now
    let k be Nat;
    DCS.(i+k+1) = DIJK:Step(DCS.(i+k)) by Def11;
    then
A2: dom (DCS.(i+k))`1 c= dom (DCS.(i+k+1))`1 & (DCS.(i+k))`2 c= (DCS.(i+k+
    1))`2 by Th16;
    assume dDCS c= dom (DCS.(i+k))`1 & eDCS c= (DCS.(i+k))`2;
    hence dDCS c= dom (DCS.(i+(k+1)))`1 & eDCS c= (DCS.(i+(k+1)))`2 by A2;
  end;
  then
A3: for k being Nat st P[k] holds P[k+1];
A4: P[ 0 ];
  for k being Nat holds P[k] from NAT_1:sch 2(A4,A3);
  hence thesis by A1;
end;
