
theorem Th2:
  for L1,L2 being non empty LattStr st the LattStr of L1 = the
LattStr of L2 for a being Element of L1, b being Element of L2, X being set st
a = b holds (a is_less_than X iff b is_less_than X) & (a is_greater_than X iff
  b is_greater_than X)
proof
  let L1,L2 be non empty LattStr such that
A1: the LattStr of L1 = the LattStr of L2;
  let a be Element of L1, b be Element of L2, X be set such that
A2: a = b;
  thus a is_less_than X implies b is_less_than X
  proof
    assume
A3: for c being Element of L1 st c in X holds a [= c;
    let c be Element of L2;
    reconsider d = c as Element of L1 by A1;
    assume c in X;
    then a [= d by A3;
    hence thesis by A1,A2;
  end;
  thus b is_less_than X implies a is_less_than X
  proof
    assume
A4: for c being Element of L2 st c in X holds b [= c;
    let c be Element of L1;
    reconsider d = c as Element of L2 by A1;
    assume c in X;
    then b [= d by A4;
    hence thesis by A1,A2;
  end;
  thus a is_greater_than X implies b is_greater_than X
  proof
    assume
A5: for c being Element of L1 st c in X holds c [= a;
    let c be Element of L2;
    reconsider d = c as Element of L1 by A1;
    assume c in X;
    then d [= a by A5;
    hence thesis by A1,A2;
  end;
  assume
A6: for c being Element of L2 st c in X holds c [= b;
  let c be Element of L1;
  reconsider d = c as Element of L2 by A1;
  assume c in X;
  then d [= b by A6;
  hence thesis by A1,A2;
end;
