reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve K,K1,K2,K3 for Field;
reserve SK1,SK2 for Subfield of K;
reserve ek,ek1,ek2 for Element of K;
reserve p for Prime;
reserve a,b,c for Element of GF(p);
reserve F for FinSequence of GF(p);
reserve Px,Py,Pz for Element of GF(p);

theorem
  for X be non empty finite set,
  R be Equivalence_Relation of X,
  S be Class(R)-valued Function,
  L be FinSequence of NAT
  st S is one-to-one onto & dom S = dom L
  & (for i be Nat st i in dom S holds L.i = card (S.i))
  holds card (X) = Sum L
  proof
    let X be non empty finite set,
    R be Equivalence_Relation of X,
    S be Class(R)-valued Function,
    L be FinSequence of NAT;
    assume
A1: S is one-to-one onto &
    dom S = dom L & (for i be Nat st i in dom S holds L.i = card (S.i));
A2: S is disjoint_valued by A1,Th55;
A3: for i be Nat st i in dom S holds
    S.i is finite & L.i = card (S.i) by A1,Th54;
    Union S = X by Th56,A1;
    hence thesis by A1,A2,A3,DIST_1:17;
  end;
