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 Th102:
  for G being _Graph, X being set, e,y being object st e SJoins X,{y},G
  ex x being object st x in X & e Joins x,y,G
proof
  let G be _Graph, X be set, e,y be object;
  assume A1: e SJoins X,{y},G;
  then A2: e in the_Edges_of G;
  per cases by A1;
  suppose A3: (the_Source_of G).e in X & (the_Target_of G).e in {y};
    take (the_Source_of G).e;
    (the_Target_of G).e = y by A3, TARSKI:def 1;
    then e DJoins (the_Source_of G).e,y,G by A2;
    hence thesis by A3;
  end;
  suppose A4: (the_Source_of G).e in {y} & (the_Target_of G).e in X;
    take (the_Target_of G).e;
    (the_Source_of G).e = y by A4, TARSKI:def 1;
    then e DJoins y,(the_Target_of G).e,G by A2;
    hence thesis by A4;
  end;
end;
