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 Th31:
  card dom A = 1 implies A = VERUM or ex p st A = @p
proof
  assume card dom A = 1;
  then consider x being object such that
A1: dom A = {x} by CARD_2:42;
  reconsider x as Element of dom A by A1,TARSKI:def 1;
A2: {} in dom A by TREES_1:22;
  then
A3: dom A = elementary_tree 0 by A1,TARSKI:def 1,TREES_1:29;
A4: dom A = {{}} by A2,A1,TARSKI:def 1;
  succ x = {}
  proof
    set y = the Element of succ x;
    assume not thesis;
    then
A5: y in succ x;
    succ x = { x^<*n*>: x^<*n*> in dom A } by TREES_2:def 5;
    then ex n st y = x^<*n*> & x^<*n*> in dom A by A5;
    hence contradiction by A4,TARSKI:def 1;
  end;
  then
A6: branchdeg x = 0 by CARD_1:27,TREES_2:def 12;
  now
    per cases by A6,Def5;
    suppose
      A.x = [0,0];
      then for z being object holds z in dom A implies A.z = [0,0]
        by A1,TARSKI:def 1;
      hence thesis by A3,FUNCOP_1:11;
    end;
    suppose
      ex n st A.x = [3,n];
      then consider n such that
A7:   A.x = [3,n];
      3 in NAT & n in NAT by ORDINAL1:def 12;
      then reconsider p = [3,n] as MP-variable by ZFMISC_1:105;
      for z being object holds z in dom A implies A.z = [3,n]
               by A1,A7,TARSKI:def 1;
      then A = @p by A3,FUNCOP_1:11;
      hence thesis;
    end;
  end;
  hence thesis;
end;
