reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;
reserve f,g for Function;
reserve A,B,C for array;
reserve O for connected non empty Poset;
reserve R,Q for array of O;
reserve T for non empty array of O;
reserve p,q,r,s for Element of dom T;
reserve A for array, B for permutation of A;

theorem
Th84: for a being finite Ordinal, x st x in a
      holds x = 0 or ex b st x = succ b
   proof
     let a be finite Ordinal;
     let x;
     assume
A1:  x in a & x <> 0; then
A2:  {} in x by ORDINAL3:8;
     now assume x is limit_ordinal; then
       omega c= x & x c= a by A1,A2,ORDINAL1:def 2,def 11;
       hence contradiction;
     end;
     hence thesis by A1,ORDINAL1:29;
   end;
