reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;
reserve e,u for object,
  A for Subset of X;

theorem Th110: :: WELLORD2:28
 for M being set
   st for X st X in M holds X <> {}
  ex f being Function
  st dom f = M & for X st X in M holds f.X in X
proof let M be set;
 assume
A1: for X st X in M holds X <> {};
  deffunc F(set) = the Element of $1;
  consider f being Function such that
A2: dom f = M and
A3: for X st X in M holds f.X = F(X) from LambdaS;
 take f;
 thus dom f = M by A2;
 let X;
 assume
A4: X in M;
  then
A5:  f.X = the Element of X by A3;
  X <> {} by A1,A4;
 hence f.X in X by A5;
end;
