reserve Y for non empty set,
  a for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  P,Q for a_partition of Y;
reserve x,y,z for set,
  S, X for non empty set,
  R for Relation of X;

theorem
  op1 = {[{},{}]}
proof
  {{}} = dom op1 by FUNCT_2:def 1;
  then {} in dom op1 by TARSKI:def 1;
  then [{},op1.{}] in op1 by FUNCT_1:def 2;
  then
A1: {[{},op1.{}]} c= op1 by ZFMISC_1:31;
  rng op1 = {op1.{}} by FUNCT_2:48;
  then
A2: op1.{} = {} by ZFMISC_1:18;
  op1 c= [:{{}},{{}}:];
  then op1 c= {[{},{}]} by ZFMISC_1:29;
  hence thesis by A2,A1;
end;
