
theorem Th19:
  for G being _finite _Graph, S being VNumberingSeq of G, m,n being
  Nat, v being set st v in dom (S.m) & v in dom (S.n) holds (S.m).v = (S.n).v
proof
  let G be _finite _Graph, S be VNumberingSeq of G, m,n be Nat;
  let v be set such that
A1: v in dom (S.m) and
A2: v in dom (S.n);
  set VLM = S.m;
A3: [v,VLM.v] in VLM by A1,FUNCT_1:def 2;
  set VLN = S.n;
A4: [v,VLN.v] in VLN by A2,FUNCT_1:def 2;
  per cases;
  suppose
    m <= n;
    then VLM c= VLN by Th17;
    hence thesis by A2,A3,FUNCT_1:def 2;
  end;
  suppose
    m > n;
    then VLN c= VLM by Th17;
    hence thesis by A1,A4,FUNCT_1:def 2;
  end;
end;
