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;
reserve x for object;

theorem
  x in X /\ Y implies In(x,X) = In(x,Y)
proof
  assume
A1: x in X /\ Y;
  then
A2: x in Y by XBOOLE_0:def 4;
  x in X by A1,XBOOLE_0:def 4;
  hence In (x,X) = x by Def7
    .= In (x,Y) by A2,Def7;
end;
