
theorem Th27:
  for L1,L2 be up-complete non empty Poset for f be Function of
  L1,L2 st f is isomorphic for x,y be Element of L1 holds x << y iff f.x << f.y
proof
  let L1,L2 be up-complete non empty Poset;
  let f be Function of L1,L2;
  assume
A1: f is isomorphic;
  then reconsider g = f" as Function of L2,L1 by WAYBEL_0:67;
  let x,y be Element of L1;
  thus x << y implies f.x << f.y by A1,Lm4;
  thus f.x << f.y implies x << y
  proof
    y in the carrier of L1;
    then
A2: y in dom f by FUNCT_2:def 1;
    x in the carrier of L1;
    then
A3: x in dom f by FUNCT_2:def 1;
    assume f.x << f.y;
    then g.(f.x) << g.(f.y) by A1,Lm4,WAYBEL_0:68;
    then x << g.(f.y) by A1,A3,FUNCT_1:34;
    hence thesis by A1,A2,FUNCT_1:34;
  end;
end;
