
theorem Th18:
  for L being non empty RelStr, N being non empty NetStr over L
  for x being Element of L holds (x"/\")*N = x "/\" N
proof
  let L be non empty RelStr, N be non empty NetStr over L, x be Element of L;
  set n = the mapping of N;
A1: the RelStr of (x"/\")*N = the RelStr of N by Def8
    .= the RelStr of x "/\" N by WAYBEL_2:def 3;
A2: the RelStr of N = the RelStr of x "/\" N by WAYBEL_2:def 3;
  then reconsider
  f2 = the mapping of x "/\" N as Function of the carrier of N, the
  carrier of L;
A3: for c being Element of N holds ((x"/\") * n).c = f2.c
  proof
    let c be Element of N;
    consider y being Element of L such that
A4: y = n.c and
A5: f2.c = x "/\" y by A2,WAYBEL_2:def 3;
    thus ((x"/\") * n).c = (x"/\").y by A4,FUNCT_2:15
      .= f2.c by A5,WAYBEL_1:def 18;
  end;
  the mapping of (x"/\")*N = (x"/\") * n by Def8
    .= the mapping of x "/\" N by A3,FUNCT_2:63;
  hence thesis by A1;
end;
