
theorem Th50: :: Adjacent01
  for G1,G2 being _Graph st G1 == G2 for S being set holds G1
  .AdjacentSet(S) = G2.AdjacentSet(S)
proof
  let G1,G2 be _Graph such that
A1: G1 == G2;
  let S be set;
A2: now
    let x be object such that
A3: x in G2.AdjacentSet(S);
    reconsider t2 = x as Vertex of G2 by A3;
A4: not t2 in S by A3,Th49;
    consider v2 being Vertex of G2 such that
A5: v2 in S and
A6: t2,v2 are_adjacent by A3,Th49;
    reconsider t1 = t2, v1 = v2 as Vertex of G1 by A1;
    t1,v1 are_adjacent by A1,A6,Th43;
    hence x in G1.AdjacentSet(S) by A4,A5;
  end;
  now
    let x be object such that
A7: x in G1.AdjacentSet(S);
    reconsider t1 = x as Vertex of G1 by A7;
A8: not t1 in S by A7,Th49;
    consider v1 being Vertex of G1 such that
A9: v1 in S and
A10: t1,v1 are_adjacent by A7,Th49;
    reconsider t2 = t1, v2 = v1 as Vertex of G2 by A1;
    t2,v2 are_adjacent by A1,A10,Th43;
    hence x in G2.AdjacentSet(S) by A8,A9;
  end;
  hence thesis by A2,TARSKI:2;
end;
