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 Th6:
  X c= M implies card X c= M
proof
  defpred P[Ordinal] means $1 c= M & X,$1 are_equipotent;
  reconsider m = M as Ordinal;
  assume X c= M;
  then
A1: order_type_of RelIncl X c= m & RelIncl X is well-ordering by WELLORD2:8,14;
  field RelIncl X = X by WELLORD2:def 1;
  then
A2: ex A st P[A] by A1,Th5;
  consider A such that
A3: P[A] & for B st P[B] holds A c= B from ORDINAL1:sch 1(A2);
  A is cardinal
  proof
    take A;
    thus A = A;
    let B;
    assume
A4: B,A are_equipotent;
    assume
A5: not A c= B;
    then m c= B by A3,A4,WELLORD2:15;
    hence contradiction by A3,A5;
  end;
  then reconsider A as Cardinal;
  card X = A by A3,Def2;
  hence thesis by A3;
end;
