
theorem Th25:
  for G being _Graph holds G is locally-finite iff
    for v being Vertex of G
    holds v.edgesIn() is finite & v.edgesOut() is finite
proof
  let G be _Graph;
  hereby
    assume A1: G is locally-finite;
    let v be Vertex of G;
    v.edgesInOut() is finite by A1;
    hence v.edgesIn() is finite & v.edgesOut() is finite;
  end;
  assume A2: for v being Vertex of G holds
    v.edgesIn() is finite & v.edgesOut() is finite;
  let v be Vertex of G;
  v.edgesIn() is finite & v.edgesOut() is finite by A2;
  hence v.edgesInOut() is finite;
end;
