
theorem Th18:
  for L1,L2 be non empty RelStr for X be Subset of L1, x be
Element of L1 for f be Function of L1,L2 st f is isomorphic holds x is_<=_than
  X iff f.x is_<=_than f.:X
proof
  let L1,L2 be non empty RelStr;
  let X be Subset of L1;
  let x be Element of L1;
  let f be Function of L1,L2;
  assume
A1: f is isomorphic;
  hence x is_<=_than X implies f.x is_<=_than f.:X by YELLOW_2:13;
  thus f.x is_<=_than f.:X implies x is_<=_than X
  proof
    reconsider g = f" as Function of L2,L1 by A1,WAYBEL_0:67;
    assume
A2: f.x is_<=_than f.:X;
    g is isomorphic by A1,WAYBEL_0:68;
    then
A3: g.(f.x) is_<=_than g.:(f.:X) by A2,YELLOW_2:13;
A4: f"(f.:X) c= X by A1,FUNCT_1:82;
    X c= the carrier of L1;
    then X c= dom f by FUNCT_2:def 1;
    then
A5: X c= f"(f.:X) by FUNCT_1:76;
    x in the carrier of L1;
    then
A6: x in dom f by FUNCT_2:def 1;
    g.:(f.:X) = f"(f.:X) by A1,FUNCT_1:85
      .= X by A4,A5,XBOOLE_0:def 10;
    hence thesis by A1,A6,A3,FUNCT_1:34;
  end;
end;
