reserve A for RelStr;
reserve X for non empty set;
reserve PX,PY,PZ,Y,a,b,c,x,y for set;
reserve S1,S2 for Subset of Y;

theorem
  for H being covering Hierarchy of Y for B being mutually-disjoint
Subset-Family of Y st B c= H & for C being mutually-disjoint Subset-Family of Y
  st C c= H & union B c= union C holds B = C holds B is a_partition of Y
proof
  let H be covering Hierarchy of Y;
  let B be mutually-disjoint Subset-Family of Y such that
A1: B c= H and
A2: for C being mutually-disjoint Subset-Family of Y st C c= H & union B
  c= union C holds B = C;
  thus union B = Y by A1,A2,Lm3;
  thus thesis by A1,A2,Def5,Lm4;
end;
