reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;
reserve e,u,v for object, E,X,Y,X1 for set;
reserve X,Y,Z for non empty set;

theorem
  for D being a_partition of X, A being Subset of D, B being
  Subset of X st B = union A holds B` = union A`
proof
  let D be a_partition of X, A be Subset of D, B be Subset of X;
  assume
A1: B = union A;
  union D = X by Def4;
  hence B` c= union A` by A1,Th55;
  let e be object;
  assume e in union A`;
  then consider u being set such that
A2: e in u and
A3: u in A` by TARSKI:def 4;
A4: u in D by A3;
  assume not e in B`;
  then e in B by A2,A4,SUBSET_1:29;
  then consider v being set such that
A5: e in v and
A6: v in A by A1,TARSKI:def 4;
A7: v in D by A6;
  not u misses v by A2,A5,XBOOLE_0:3;
  then u = v by A4,A7,Def4;
  hence contradiction by A3,A6,XBOOLE_0:def 5;
end;
