reserve k,m,n for Element of NAT,
  a,X,Y for set,
  D,D1,D2 for non empty set;
reserve p,q for FinSequence of NAT;
reserve x,y,z,t for Variable;
reserve F,F1,G,G1,H,H1 for ZF-formula;

theorem Th13:
  3 <= len H
proof
  now
    assume not H is atomic;
    then consider H1 such that
A1: len H1 + 1 <= len H by Th12;
A2: now
      assume not H1 is atomic;
      then consider F such that
A3:   len F + 1 <= len H1 by Th12;
A4:   now
        assume not F is atomic;
        then consider F1 such that
A5:     len F1 + 1 <= len F by Th12;
        0 + 1 <= len F1 + 1 by XREAL_1:7;
        then 1 <= len F by A5,XXREAL_0:2;
        then 1 + 1 <= len F + 1 by XREAL_1:7;
        then 2 <= len H1 by A3,XXREAL_0:2;
        then 2 + 1 <= len H1 + 1 by XREAL_1:7;
        hence thesis by A1,XXREAL_0:2;
      end;
A6:   len F + 1 + 1 <= len H1 + 1 by A3,XREAL_1:7;
      now
        assume F is atomic;
        then len F = 3 by Th11;
        then 3 + 1 + 1 <= len H by A1,A6,XXREAL_0:2;
        hence thesis by XXREAL_0:2;
      end;
      hence thesis by A4;
    end;
    now
      assume H1 is atomic;
      then 3 + 1 <= len H by A1,Th11;
      hence thesis by XXREAL_0:2;
    end;
    hence thesis by A2;
  end;
  hence thesis by Th11;
end;
