
theorem Th11:
  for A being non empty set for L being lower-bounded LATTICE for
  d being BiFunction of A,L st d is symmetric for q being Element of [:A,A,the
  carrier of L,the carrier of L:] holds new_bi_fun2(d,q) is symmetric
proof
  let A be non empty set;
  let L be lower-bounded LATTICE;
  let d be BiFunction of A,L;
  assume
A1: d is symmetric;
  let q be Element of [:A,A,the carrier of L,the carrier of L:];
  set f = new_bi_fun2(d,q), x = q`1_4, y = q`2_4, a = q`3_4, b = q`4_4;
  let p,q be Element of new_set2 A;
A2: p in A or p in {{A},{{A}}} by XBOOLE_0:def 3;
A3: q in A or q in {{A},{{A}}} by XBOOLE_0:def 3;
  per cases by A2,A3,TARSKI:def 2;
  suppose
    p in A & q in A;
    then reconsider p9 = p, q9 = q as Element of A;
    thus f.(p,q) = d.(p9,q9) by Def4
      .= d.(q9,p9) by A1
      .= f.(q,p) by Def4;
  end;
  suppose
A4: p in A & q = {A};
    then reconsider p9 = p as Element of A;
    thus f.(p,q) = d.(p9,x)"\/"a by A4,Def4
      .= f.(q,p) by A4,Def4;
  end;
  suppose
A5: p in A & q = {{A}};
    then reconsider p9 = p as Element of A;
    thus f.(p,q) = d.(p9,y)"\/"a by A5,Def4
      .= f.(q,p) by A5,Def4;
  end;
  suppose
A6: p = {A} & q in A;
    then reconsider q9 = q as Element of A;
    thus f.(p,q) = d.(q9,x)"\/"a by A6,Def4
      .= f.(q,p) by A6,Def4;
  end;
  suppose
    p = {A} & q = {A};
    hence thesis;
  end;
  suppose
A7: p = {A} & q = {{A}};
    hence f.(p,q) = (d.(x,y)"\/"a)"/\"b by Def4
      .= f.(q,p) by A7,Def4;
  end;
  suppose
A8: p = {{A}} & q in A;
    then reconsider q9 = q as Element of A;
    thus f.(p,q) = d.(q9,y)"\/"a by A8,Def4
      .= f.(q,p) by A8,Def4;
  end;
  suppose
A9: p = {{A}} & q = {A};
    hence f.(p,q) = (d.(x,y)"\/"a)"/\"b by Def4
      .= f.(q,p) by A9,Def4;
  end;
  suppose
    p = {{A}} & q = {{A}};
    hence thesis;
  end;
end;
