reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;
reserve n for Element of omega;
reserve e,u for set;

theorem
  for f being set, p being Relation
  for x being set st x in rng p holds the_rank_of x in the_rank_of [p,f]
proof
  let f be set;
  let p be Relation;
  let y be set;
  assume y in rng p;
  then consider x being object such that
A1: [x,y] in p by XTUPLE_0:def 13;
A2: p in {p,f} by TARSKI:def 2;
A3: {p,f} in {{p,f},{p}} by TARSKI:def 2;
A4: y in {x,y} by TARSKI:def 2;
A5: {x,y} in {{x,y},{x}} by TARSKI:def 2;
A6: the_rank_of y in the_rank_of {x,y} by A4,Th68;
A7: the_rank_of {x,y} in the_rank_of [x,y] by A5,Th68;
A8: the_rank_of p in the_rank_of {p,f} by A2,Th68;
A9: the_rank_of {p,f} in the_rank_of [p,f] by A3,Th68;
A10: the_rank_of y in the_rank_of [x,y] by A6,A7,ORDINAL1:10;
A11: the_rank_of [x,y] in the_rank_of p by A1,Th68;
A12: the_rank_of p in the_rank_of [p,f] by A8,A9,ORDINAL1:10;
 the_rank_of y in the_rank_of p by A10,A11,ORDINAL1:10;
  hence thesis by A12,ORDINAL1:10;
end;
