reserve GS for GraphStruct;
reserve G,G1,G2,G3 for _Graph;
reserve e,x,x1,x2,y,y1,y2,E,V,X,Y for set;
reserve n,n1,n2 for Nat;
reserve v,v1,v2 for Vertex of G;

theorem Th63:
  e Joins v1,v2,G implies e in v1.edgesIn() & e in v2.edgesOut()
  or e in v2.edgesIn() & e in v1.edgesOut()
proof
  assume
A1: e Joins v1,v2,G;
  then
A2: e in the_Edges_of G;
  now
    per cases by A1;
    suppose
      (the_Source_of G).e = v1 & (the_Target_of G).e = v2;
      hence thesis by A2,Lm7,Lm8;
    end;
    suppose
      (the_Source_of G).e = v2 & (the_Target_of G).e = v1;
      hence thesis by A2,Lm7,Lm8;
    end;
  end;
  hence thesis;
end;
