
theorem Th77:
  for G1 being _Graph, G2 being removeDParallelEdges of G1
  for v1,w1 being Vertex of G1, v2,w2 being Vertex of G2
  st v1 = v2 & w1 = w2 holds v1,w1 are_adjacent iff v2,w2 are_adjacent
proof
  let G1 be _Graph, G2 be removeDParallelEdges of G1;
  let v1,w1 be Vertex of G1, v2,w2 be Vertex of G2;
  assume A1: v1 = v2 & w1 = w2;
  hereby
    assume v1,w1 are_adjacent;
    then w1 in v1.allNeighbors() by Th42;
    then w2 in v2.allNeighbors() by A1, GLIBPRE0:65;
    hence v2,w2 are_adjacent by Th42;
  end;
  assume v2,w2 are_adjacent;
  then w2 in v2.allNeighbors() by Th42;
  then w1 in v1.allNeighbors() by A1, GLIBPRE0:65;
  hence v1,w1 are_adjacent by Th42;
end;
