
theorem Th75:
  for G1 being _Graph, G2 being removeLoops of G1
  for v1,w1 being Vertex of G1, v2,w2 being Vertex of G2
  st v1 = v2 & w1 = w2 & v1 <> w1
  holds v1,w1 are_adjacent iff v2,w2 are_adjacent
proof
  let G1 be _Graph, G2 be removeLoops of G1;
  let v1,w1 be Vertex of G1, v2,w2 be Vertex of G2;
  assume A1: v1 = v2 & w1 = w2 & v1 <> w1;
  hereby
    assume v1,w1 are_adjacent;
    then w1 in v1.allNeighbors() \ {v1} by A1, Th42, ZFMISC_1:56;
    then w2 in v2.allNeighbors() by A1, GLIBPRE0:63;
    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() \ {v1} by A1, GLIBPRE0:63;
  hence v1,w1 are_adjacent by Th42, ZFMISC_1:56;
end;
