reserve x for set;

theorem Th5:
  for L1, L2 being non empty 1-sorted st the carrier of L1 = the
carrier of L2 for N1 being net of L1, N2 being net of L2 st the RelStr of N1 =
the RelStr of N2 & the mapping of N1 = the mapping of N2 for S1 being subnet of
N1 ex S2 being strict subnet of N2 st the RelStr of S1 = the RelStr of S2 & the
  mapping of S1 = the mapping of S2
proof
  let L1, L2 be non empty 1-sorted such that
A1: the carrier of L1 = the carrier of L2;
  let N1 be net of L1, N2 be net of L2 such that
A2: the RelStr of N1 = the RelStr of N2 and
A3: the mapping of N1 = the mapping of N2;
  let S1 be subnet of N1;
  consider S2 being strict NetStr over L2 such that
A4: the RelStr of S1 = the RelStr of S2 and
A5: the mapping of S1 = the mapping of S2 by A1,Th2;
  reconsider S2 as strict net of L2 by A4;
  consider f being Function of S1, N1 such that
A6: the mapping of S1 = (the mapping of N1)*f and
A7: for B5 being Element of N1 holds ex B6 being Element of S1 st for B7
  being Element of S1 st B6 <= B7 holds B5 <= f.B7 by YELLOW_6:def 9;
  reconsider g = f as Function of S2, N2 by A2,A4;
  S2 is subnet of N2
  proof
    take g;
    thus the mapping of S2 = (the mapping of N2)*g by A3,A5,A6;
    let B2 be Element of N2;
    reconsider b2 = B2 as Element of N1 by A2;
    consider b6 being Element of S1 such that
A8: for b7 being Element of S1 st b6 <= b7 holds b2 <= f.b7 by A7;
    reconsider B3 = b6 as Element of S2 by A4;
    take B3;
    let B4 be Element of S2;
    reconsider b4 = B4 as Element of S1 by A4;
    assume B3 <= B4;
    then b6 <= b4 by A4,YELLOW_0:1;
    then b2 <= f.b4 by A8;
    hence thesis by A2,YELLOW_0:1;
  end;
  hence thesis by A4,A5;
end;
