reserve x,y,x1,x2,z for set,
  n,m,k for Nat,
  t1 for (DecoratedTree of [: NAT,NAT :]),
  w,s,t,u for FinSequence of NAT,
  D for non empty set;
reserve s9,w9,v9 for Element of NAT*;
reserve p,q for MP-variable;
reserve A,A1,B,B1,C,C1 for MP-wff;

theorem Th33:
  card dom A < card dom 'not' A
proof
  set e = elementary_tree 1;
  <*0*> in e by TARSKI:def 2,TREES_1:51;
  then
A1: <*0*> in dom (e --> [1,0]) by FUNCOP_1:13;
  then
A2: dom 'not' A = dom (e --> [1,0]) with-replacement (<*0*>, dom A) by
TREES_2:def 11;
  then reconsider o = <*0*> as Element of dom 'not' A by A1,TREES_1:def 9;
  now
    let s;
    thus s in dom A implies o^s in dom 'not' A by A1,A2,TREES_1:def 9;
    assume
A3: o^s in dom 'not' A;
    now
      per cases;
      suppose
        s = {};
        hence s in dom A by TREES_1:22;
      end;
      suppose
        s <> {};
        then o is_a_proper_prefix_of o^s by TREES_1:10;
        then ex w st w in dom A & o^s = o^w by A1,A2,A3,TREES_1:def 9;
        hence s in dom A by FINSEQ_1:33;
      end;
    end;
    hence s in dom A;
  end;
  then
A4: dom A = (dom 'not' A)|o by TREES_1:def 6;
  o <> Root (dom 'not' A);
  hence thesis by A4,Th16;
end;
