
theorem Th7:
  for L being lower-bounded LATTICE for x,y being Element of L for
A being non empty set for f be Homomorphism of L,EqRelLATT A st f is one-to-one
  holds (corestr f).x <= (corestr f).y implies x <= y
proof
  let L be lower-bounded LATTICE;
  let x,y be Element of L;
  let A be non empty set;
  let f be Homomorphism of L,EqRelLATT A;
  assume that
A1: f is one-to-one and
A2: (corestr f).x <= (corestr f).y;
  now
A3: corestr f = f by WAYBEL_1:30;
A4: for x,y being Element of L holds (corestr f).(x "/\" y) = (corestr f).
    x "/\" (corestr f).y
    proof
      let x,y be Element of L;
      thus (corestr f).(x "/\" y) = f.x "/\" f.y by A3,WAYBEL_6:1
        .= (corestr f).x "/\" (corestr f).y by A3,YELLOW_0:69;
    end;
A5: corestr f is one-to-one by A1,WAYBEL_1:30;
    (corestr f).y "/\" (corestr f).x = (corestr f).x by A2,YELLOW_5:10;
    then (corestr f).x = (corestr f).(x "/\" y) by A4;
    then
A6: x = x "/\" y by A5,WAYBEL_1:def 1;
    assume not x <= y;
    hence contradiction by A6,YELLOW_0:25;
  end;
  hence thesis;
end;
