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;

theorem Th15:
  card X = card Y implies nextcard X = nextcard Y
proof
  assume
A1: card X = card Y;
  card X in nextcard X & for N st card X in N holds nextcard X c= N by Def3;
  hence thesis by A1,Def3;
end;
