
theorem Th5:
  for G being real-weighted WGraph, EL being FF:ELabeling of G,
  source being Vertex of G, i,j being Nat st i <= j holds dom (AP:CompSeq(EL,
  source).i) c= dom (AP:CompSeq(EL,source).j)
proof
  let G be real-weighted WGraph, EL be FF:ELabeling of G, source be Vertex of
  G, i,j be Nat;
  set CS = AP:CompSeq(EL,source);
  defpred P[Nat] means dom (CS.i) c= dom (CS.(i+$1));
  assume i <= j;
  then consider k being Nat such that
A1: j = i+k by NAT_1:10;
A2: now
    let n be Nat;
    set Gn = (CS.(i+n)), Gn1 = (CS.(i+(n+1)));
    set Next = AP:NextBestEdges(Gn), e = the Element of Next;
    Gn1 = (CS.(i+n+1));
    then
A3: Gn1 = AP:Step(Gn) by Def12;
    assume
A4: P[n];
    now
      per cases;
      suppose
        Next = {};
        hence P[n+1] by A4,A3,Def10;
      end;
      suppose
        Next <> {} & not (the_Source_of G).e in dom Gn;
        then Gn1 = Gn+*((the_Source_of G).e .--> e) by A3,Def10;
        then dom Gn c= dom Gn1 by FUNCT_4:10;
        hence P[n+1] by A4,XBOOLE_1:1;
      end;
      suppose
        Next <> {} & (the_Source_of G).e in dom Gn;
        then Gn1 = Gn+*((the_Target_of G).e .--> e) by A3,Def10;
        then dom Gn c= dom Gn1 by FUNCT_4:10;
        hence P[n+1] by A4,XBOOLE_1:1;
      end;
    end;
    hence P[n+1];
  end;
A5: P[ 0 ];
A6: for n being Nat holds P[n] from NAT_1:sch 2(A5,A2);
  thus thesis by A6,A1;
end;
