theorem Th61:
  e in v.edgesInOut() iff e in the_Edges_of G & ((the_Source_of G)
  .e = v or (the_Target_of G).e = v)
proof
  hereby
    assume
A1: e in v.edgesInOut();
    hence e in the_Edges_of G;
    e in v.edgesIn() or e in v.edgesOut() by A1,XBOOLE_0:def 3;
    hence (the_Source_of G).e=v or (the_Target_of G).e=v by Lm7,Lm8;
  end;
  assume
  e in the_Edges_of G &( (the_Source_of G).e=v or (the_Target_of G).e= v);
  then e in v.edgesIn() or e in v.edgesOut() by Lm7,Lm8;
  hence thesis by XBOOLE_0:def 3;
end;
