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 non empty a_partition of X, W being Subset of X st
for B being Subset of X st B in D & B meets W holds B c= W holds W = proj D " (
  proj D .: W)
proof
  let D be non empty a_partition of X, W be Subset of X such that
A1: for B being Subset of X st B in D & B meets W holds B c= W;
  W c= X;
  then W c= dom proj D by FUNCT_2:def 1;
  hence W c= proj D " (proj D .: W) by FUNCT_1:76;
  let e be object;
  assume
A2: e in proj D " (proj D .: W);
  then reconsider d = e as Element of X;
  (proj D).e in proj D .: W by A2,FUNCT_1:def 7;
  then consider c being Element of X such that
A3: c in W and
A4: (proj D).d = (proj D).c by FUNCT_2:65;
  reconsider B = (proj D).c as Subset of X by TARSKI:def 3;
  c in (proj D).c by Def9;
  then B meets W by A3,XBOOLE_0:3;
  then
A5: B c= W by A1;
  d in B by A4,Def9;
  hence thesis by A5;
end;
