
theorem Th19:
  for G1,G2 being WGraph, W1 being Walk of G1, W2 being Walk of G2
  st W1 = W2 & the_Weight_of G1 = the_Weight_of G2 holds W1.weightSeq() = W2
  .weightSeq()
proof
  let G1,G2 be WGraph, W1 be Walk of G1, W2 be Walk of G2;
  assume that
A1: W1 = W2 and
A2: the_Weight_of G1 = the_Weight_of G2;
  set WS1 = W1.weightSeq(), WS2 = W2.weightSeq();
A3: W1.edgeSeq() = W2.edgeSeq() by A1,GLIB_001:86;
  now
    thus len WS1 = len WS1;
    thus
A4: len WS2 = len W1.edgeSeq() by A3,Def18
      .= len WS1 by Def18;
    let x be Nat;
    assume
A5: x in dom WS1;
    then
A6: 1 <= x by FINSEQ_3:25;
A7: x <= len WS1 by A5,FINSEQ_3:25;
    x <= len WS2 by A4,A5,FINSEQ_3:25;
    hence WS2.x = (the_Weight_of G2).(W2.edgeSeq().x) by A6,Def18
      .= (the_Weight_of G1).(W1.edgeSeq().x) by A1,A2,GLIB_001:86
      .= WS1.x by A6,A7,Def18;
  end;
  hence thesis by FINSEQ_2:9;
end;
