reserve x for set;

theorem Th7:
  for L1, L2 being non empty 1-sorted for N1 being non empty NetStr
over L1 for N2 being non empty NetStr over L2 st the RelStr of N1 = the RelStr
  of N2 & the mapping of N1 = the mapping of N2 for X being set st N1
  is_eventually_in X holds N2 is_eventually_in X
proof
  let L1, L2 be non empty 1-sorted;
  let N1 be non empty NetStr over L1;
  let N2 be non empty NetStr over L2 such that
A1: the RelStr of N1 = the RelStr of N2 and
A2: the mapping of N1 = the mapping of N2;
  let X be set;
  given i1 being Element of N1 such that
A3: for j being Element of N1 st i1 <= j holds N1.j in X;
  reconsider i2 = i1 as Element of N2 by A1;
  take i2;
  let j2 be Element of N2;
  reconsider j1 = j2 as Element of N1 by A1;
  assume i2 <= j2;
  then N1.j1 in X by A1,A3,YELLOW_0:1;
  hence thesis by A2;
end;
