
theorem th0k:
for n,m being Nat holds (n c= m iff n <= m) & (n in m iff n < m)
proof
let n,m be Nat;
H: n = Segm n & m = Segm m;
A: now assume n c= m;
   then card Segm n c= card Segm m;
   hence n <= m by NAT_1:40;
   end;
now assume n <= m;
  then card Segm n c= card Segm m by NAT_1:40;
  hence n c= m;
  end;
hence n c= m iff n <= m by A;
A: now assume n in m;
   then card n in card m;
   hence n < m by H,NAT_1:41;
   end;
now assume n < m;
  then card Segm n in card Segm m by NAT_1:41;
  hence n in m;
  end;
hence n in m iff n < m by A;
end;
