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 Th57:
  e in v.edgesIn() iff ex x being set st e DJoins x,v,G
proof
  hereby
    set x = (the_Source_of G).e;
    assume
A1: e in v.edgesIn();
    take x;
    (the_Target_of G).e = v by A1,Lm7;
    hence e DJoins x,v,G by A1;
  end;
  given x being set such that
A2: e DJoins x,v,G;
  e in the_Edges_of G & (the_Target_of G).e = v by A2;
  hence thesis by Lm7;
end;
