reserve E,X,Y,x for set;
reserve A,B,C for Subset of E;
reserve x1,x2,x3,x4,x5,x6,x7,x8,x9,x10 for Element of X;

theorem
  for X being non empty set, A, B being non empty Subset of X st A c< B
   ex p being Element of X st p in B & A c= B \ {p}
proof
  let X be non empty set, A, B be non empty Subset of X;
  assume
A1: A c< B;
  then consider p being Element of X such that
A2: p in B \ A by Th4,XBOOLE_1:105;
A3: not p in A by A2,XBOOLE_0:def 5;
  take p;
  thus thesis by A1,A2,A3,XBOOLE_0:def 5,ZFMISC_1:34;
end;
