
theorem Th34:
  for I being 2-element set, f being ManySortedSet of I
  for i, j being Element of I, x being object st i <> j
  holds f +* (i,x) = (i,j) --> (x,f.j) & f +* (j,x) = (i,j) --> (f.i,x)
proof
  let I be 2-element set, f be ManySortedSet of I;
  let i, j be Element of I, x be object;
  assume A1: i <> j;
  thus f +* (i,x) = (i,j) --> (x,f.j) by A1, Lm2;
  thus f +* (j,x) = (j,i) --> (x,f.i) by A1, Lm2
    .= (i,j) --> (f.i,x) by A1, Th10;
end;
