
theorem Th34:
  for G being _finite real-weighted WGraph, i,j being Nat st i <= j
holds (PRIM:CompSeq(G).i)`1 c= (PRIM:CompSeq(G).j)`1 & (PRIM:CompSeq(G).i)`2 c=
  (PRIM:CompSeq(G).j)`2
proof
  let G be _finite real-weighted WGraph, i,j be Nat;
  set PCS = PRIM:CompSeq(G);
  set vPCS = (PCS.i)`1, ePCS = (PCS.i)`2;
  defpred P[Nat] means vPCS c= (PCS.(i+$1))`1 & ePCS c= (PCS.(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;
    PCS.(i+k+1) = PRIM:Step(PCS.(i+k)) by Def17;
    then
A2: (PCS.(i+k))`1 c= (PCS.(i+k+1))`1 & (PCS.(i+k))`2 c= (PCS.(i+k+1))`2 by Th29
;
    assume vPCS c= (PCS.(i+k))`1 & ePCS c= (PCS.(i+k))`2;
    hence vPCS c= (PCS.(i+(k+1)))`1 & ePCS c= (PCS.(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;
