reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;

theorem Th76:
  for G2, G1 for e,v1,v2 being object st e Joins v1,v2,G1
    holds e Joins v1,v2,G2 or not e in the_Edges_of G2
proof
  let G2, G1;
  let e,v1,v2 be object;
  assume e Joins v1,v2,G1;
  then per cases by GLIB_000:16;
  suppose e DJoins v1,v2,G1;
    then per cases by Th75;
    suppose e DJoins v1,v2,G2;
      hence thesis by GLIB_000:16;
    end;
    suppose not e in the_Edges_of G2;
      hence thesis;
    end;
  end;
  suppose e DJoins v2,v1,G1;
    then per cases by Th75;
    suppose e DJoins v2,v1,G2;
      hence thesis by GLIB_000:16;
    end;
    suppose not e in the_Edges_of G2;
      hence thesis;
    end;
  end;
end;
