reserve m for Cardinal,
  A,B,C for Ordinal,
  x,y,z,X,Y,Z,W for set,
  f for Function;
reserve f,g for Function,
  L for Sequence,
  F for Cardinal-Function;

theorem Th41:
  X is epsilon-transitive implies card the_rank_of X c= card X
proof
  consider f such that
A1: dom f = X &
for x being object st x in X holds f.x = f(x) from FUNCT_1:sch 3;
  assume
A2: X is epsilon-transitive;
  the_rank_of X c= rng f
  proof
    let x be object;
    assume
A3: x in the_rank_of X;
    then reconsider x9 = x as Ordinal;
    consider Y such that
A4: Y in X and
A5: the_rank_of Y = x9 by A2,A3,Th40;
    f.Y = x by A1,A4,A5;
    hence thesis by A1,A4,FUNCT_1:def 3;
  end;
  hence thesis by A1,CARD_1:12;
end;
