reserve X for set, R,R1,R2 for Relation;
reserve x,y,z for set;
reserve n,m,k for Nat;

theorem
  for X being finite set
  for O being Operation of X, x,y being Element of X holds
  x,y in value_of number_of O iff card(x.O) < card(y.O)
  proof
    let X be finite set;
    let O be Operation of X;
    let x,y be Element of X;
    hereby
      assume x,y in value_of number_of O; then
      (number_of O).x < (number_of O).y by Def5; then
      card(x.O) < (number_of O).y by Def6;
      hence card(x.O) < card(y.O) by Def6;
    end;
    assume A1: card(x.O) < card(y.O);
    0 <= card(x.O) by NAT_1:2; then
    y.O <> {} by A1; then
    y in dom O & (number_of O).x = card(x.O) & (number_of O).y = card(y.O)
    by Def6,RELAT_1:170;
    hence x,y in value_of number_of O by A1,Def5;
  end;
