reserve A,B,C for Ordinal,
  X,X1,Y,Y1,Z for set,a,b,b1,b2,x,y,z for object,
  R for Relation,
  f,g,h for Function,
  k,m,n for Nat;
reserve M,N for Cardinal;
reserve S for Sequence;

theorem Th36:
  x in omega implies x is cardinal
proof
  assume that
A1: x in omega and
A2: for B st x = B ex C st C,B are_equipotent & not B c= C;
  reconsider A = x as Ordinal by A1;
  consider B such that
A3: B,A are_equipotent and
A4: not A c= B by A2;
  B in A by A4,ORDINAL1:16;
  then B in omega by A1,ORDINAL1:10;
  then reconsider n = A, m = B as Nat by A1;
  n,m are_equipotent by A3;
  hence contradiction by A4,Lm2;
end;
