reserve x,y,X for set;

theorem Th7:
  for S being non empty 1-sorted, N being net of S for i being
  Element of N, x being set holds x in rng the mapping of N|i iff ex j being
  Element of N st i <= j & x = N.j
proof
  let S be non empty 1-sorted, N be net of S;
  let i be Element of N, x be set;
A1: dom the mapping of N|i = the carrier of N|i by FUNCT_2:def 1;
  hereby
    assume x in rng the mapping of N|i;
    then consider y being object such that
A2: y in the carrier of N|i and
A3: x = (the mapping of N|i).y by A1,FUNCT_1:def 3;
    reconsider y as Element of N|i by A2;
    consider j being Element of N such that
A4: j = y and
A5: i <= j by WAYBEL_9:def 7;
    take j;
    thus i <= j by A5;
    thus x = (N|i).y by A3
      .= N.j by A4,WAYBEL_9:16;
  end;
  given j being Element of N such that
A6: i <= j and
A7: x = N.j;
  reconsider k = j as Element of N|i by A6,WAYBEL_9:def 7;
A8: x = (N|i).k by A7,WAYBEL_9:16
    .= (the mapping of N|i).j;
  j in the carrier of N|i by A6,WAYBEL_9:def 7;
  hence thesis by A1,A8,FUNCT_1:def 3;
end;
