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 Th35:
  card dom A < card dom(A '&' B) & card dom B < card dom (A '&' B)
proof
  set e = elementary_tree 2;
  set y = (e-->[2,0]) with-replacement(<*0*>,A);
A1: not <*1*> is_a_proper_prefix_of <*0*> by TREES_1:52;
A2: <*0*> in e & dom (e --> [2,0]) = e by FUNCOP_1:13,TREES_1:28;
  then
A3: dom y = dom(e-->[2,0]) with-replacement (<*0*>,dom A) by TREES_2:def 11;
  <*1*> in e & not <*0*> is_a_proper_prefix_of <*1*> by TREES_1:28,52;
  then
A4: <*1*> in dom y by A2,A3,TREES_1:def 9;
  then
A5: dom (A '&' B) = dom y with-replacement (<*1*>,dom B) by TREES_2:def 11;
  then reconsider u = <*1*> as Element of dom(A '&' B) by A4,TREES_1:def 9;
  <*0*> in dom y by A2,A3,TREES_1:def 9;
  then reconsider o = <*0*> as Element of dom(A '&' B) by A4,A5,A1,
TREES_1:def 9;
  now
    let s;
    thus s in dom A implies o^s in dom(A '&' B)
    proof
      assume s in dom A;
      then
A6:   o^s in dom y by A2,A3,TREES_1:def 9;
      not <*1*> is_a_proper_prefix_of o^s by Th2;
      hence thesis by A4,A5,A6,TREES_1:def 9;
    end;
    assume
A7: o^s in dom(A '&' B);
    now
      per cases;
      suppose
        s = {};
        hence s in dom A by TREES_1:22;
      end;
      suppose
A8:     s <> {};
        not ex w st w in dom B & o^s = <*1*>^w by TREES_1:1,50;
        then
A9:    o^s in dom y by A4,A5,A7,TREES_1:def 9;
        o is_a_proper_prefix_of o^s by A8,TREES_1:10;
        then ex w st w in dom A & o^s = o^w by A2,A3,A9,TREES_1:def 9;
        hence s in dom A by FINSEQ_1:33;
      end;
    end;
    hence s in dom A;
  end;
  then
A10: dom A = (dom(A '&' B))|o by TREES_1:def 6;
  now
    let s;
    thus s in dom B implies u^s in dom(A '&' B) by A4,A5,TREES_1:def 9;
    assume
A11: u^s in dom(A '&' B);
    now
      per cases;
      suppose
        s = {};
        hence s in dom B by TREES_1:22;
      end;
      suppose
        s <> {};
        then <*1*> is_a_proper_prefix_of u^s by TREES_1:10;
        then ex w st w in dom B & u^s = <*1*>^w by A4,A5,A11,TREES_1:def 9;
        hence s in dom B by FINSEQ_1:33;
      end;
    end;
    hence s in dom B;
  end;
  then
A12: dom B = (dom(A '&' B))|u by TREES_1:def 6;
  o <> Root (dom(A '&' B));
  hence card dom A < card dom(A '&' B) by A10,Th16;
  u <> Root (dom(A '&' B));
  hence thesis by A12,Th16;
end;
